TREATISE Calculus of Variations, ARRANGED WITH THE PURPOSE OF INTRODUCING, AS WELL AS ILLUS- TRATING, ITS PRINCIPLES TO THE READER BY MEANS OF PROBLEMS, AND DESIGNED TO PRESENT IN ALL IMPORTANT PARTICULARS A COMPLETE VIEW OF THE PRESENT STATE OF THE SCIENCE. BY LEWIS BUFFETT CARLL, A.M. NEW YORK: JOHN WILEY AND SONS, 53 East Tenth Street, Second door west ot Broadway. 1890. Q^ ""15 Copyright, i88i, by LEWIS BUFFETT CARLL, A PREFACE. Thirty years have now elapsed since the appearance of the treatise on the Calculus of Variations by Prof. Jellett, which, although it had been preceded by the smaller work of Woodhouse in 1810, and of Abbatt in 1837, is justly deemed the only complete treatise which has ever appeared in Eng- lish. But all the works named have long since been out of print, and are now so rare as not to be found in the majority of the college libraries of the United States. Moreover, even Prof. Jellett's treatise can no longer be regarded as complete, since its author had not read the memoirs of Sarrus and Cauchy relative to multiple integrals, while the contributions of Hesse, Moigno and Lindelof, and Todhunter were subse- quent to the publication of his work. It should be added, also, that all the memoirs and contributions just named are contained in works which are likewise out of print, and are now almost as difificult of access to the general reader as is that of Prof. Jellett. These considerations first led the author to undertake the preparation of the present treatise, in which he has endeav- ored to present, in as simple a manner as he could command, everything of importance which is at present known concern- ing this abstruse department of analysis. In the execution of this design the following method has, so far as possible, been pursued : When a new principle is to be introduced for the first time, a simple problem involving it is first proposed, and the principle is established when re- IV PREFACE. quired in the discussion of this problem. This having been followed by other problems of the same class, the general theory of the subject is finally given and illustrated by one or two of the most difficult problems obtainable; after which another principle is introduced in like manner. Although the view taken of a variation is that of Profs. Airy and Todhunter, and the methods of varying functions are those of Jellett and Strauch, still all the other leading conceptions and methods have, it is hoped, been explained with sufficient fulness to enable the reader to follow them when they occur in other works. The history of the subject is also briefly given in the last chapter, it being believed that the proper time for the presen- tation of the history of any science is after the reader has become familiar with its principles, as it can then, by the use of some technical terms, be accomplished more fully within a given space. To aid the non-classical reader, the use of Greek letters has, with the exception of two, whose use is now universal, and which are explained, been avoided, except in references, or in such passages as may be omitted without serious loss. Attention is also called to the words br ac hist oc krone and parallelepipedon, which are in this work spelled according to their derivation. The correct orthography of the former had been previously adopted by Moigno and Todhunter, and it is hoped that it may be sufficient to call the attention of Greek scholars to the latter. One of the great obstacles to the preparation of the pres- ent treatise has been the difficulty of procuring the author- ities which it w^as necessary to consult ; and the author would here return his thanks to the officers of his Alma Mater, Columbia College ; to Dr. Noah Porter, the President, and Mr. A. Van Name, the Librarian, of Yale College ; and to Mr. Walter M. Ferris, of Bay Ridge, L. I., for the extended loan of rare works which could not be found in other libraries, or PREFACE. T if found, could not be had at home for that careful study which they in many cases demanded. The author is also greatly indebted to Lieut. Fred. V. Abbot, U.S.iV. ; to M. S. Wilson, Ph.B., to Prof. P. Winter, and to the late A. San- der, Ph.D., all of the Flushing Institute, for valuable assist- ance in the examination of French and German works. But the greater part of the assistance which the author has received was rendered by his youngest brother, who, in addition to aiding in the examination of many works, recopied the manu- script for the printer, and subsequently undertook, in con- junction with the author himself, the proof-reading of the entire publication. It having been found necessary to publish the present treatise by subscription, the author, supported by President Barnard of Columbia College, Prof. J. H. Van Amringe of the same, Joseph W. Harper, Jr., and others, issued an appeal to the public, which shortly elicited the following subscriptions, the copies being placed at four dollars each : Seth Low and A. A. Low, 25 copies each. D. Appleton & Co., 12 copies. Richard L. Leggett and John Claflin, 10 copies each. A. S. Barnes & Co., 6 copies. Joseph W. Harper, Jr., Chas. Scribner's Sons, Ivison, Blake- man, Taylor & Co., F. A. P. Barnard, LL.D., Prof. J. H. Van Amringe, Columbia College Libraries, Gen. Alexander S. Webb, John H. Ireland, Malcolm Graham, Franklin B. Lord, Francis A. Stout, Fred. A. Schermerhorn, Frank D. Sturges, Robert Shepard, Edward Mitchell, E. H. Nichols, Prof. Felix Adler, W. Bayard Cutting, Hon. Benj. W. Downing, A. Ernest Vanderpoel, John Cropper, Willard Bartlett, Clarence R. Conger, Wm. Macnevan Purdy and Chas. Pratt, 5 copies each. Prof. C. W. Jones, 4 copies. Wm. C. Schermerhorn, J. Harsen Rhoades, Prof. E. M. Bass, Henry C. Sturges and Dr. Edw. L. Beadle, 3 copies each. VI PREFACE. Hon. Abram S. Hewitt, Gerard Beekman, Geo. P. Put- nam's Sons, Dr. Geo. M. Peabody, Chas. A. Silliman, Hon. Robt. Ray Hamilton, Morgan Dix, S.T.D., Wm. B. Wait, Mrs. Asa D. Lord, Dr. J. W. S. Arnold, Dr. R. W. Witthaus, Rev. Fred. B. Carter, Mrs. C. Roberts, R. L. Belknap, Prof. C. M. Nairne, J. Forsythe, D.D., R. L. Kennedy, John A. Monsell, Robt. Willets and John F. Carll, 2 copies each. Johns Hopkins University, Williams College, Dartmouth College, National College of Deaf Mutes, Perkins Institute for the Blind, Kentucky Institute for the Blind, Indiana Institute for the Blind, B. B. Huntoon, A. M. Shotwell, Henry Bogert, N. J. Gates, Prof. E. L. Youmans, Prof. Wm. G. Peck, Prof. Henry Drisler, Prof. Ogden M. Rood, Prof. Chas. Short, Rev. Spencer S. Roche, W. E. Byerly, Prof. T. H. Safford, J. P. Paulison, M. M. Backus, A. Wilkenson, J. H. Broully, Geo. H. Mussett, F. L. Nichols, Col. Chas. McK. Loeser, Prof. Samuel Hart, Prof. W. W. Beman, S. P. Nash, J. McL. Nash, O. R. Willis, Ph.D., S.Vernon Mann, Hon. Wm. H. Onderdonk, Henry Onderdonk, Rev. E. A. Dalrymple, Gouve- neur M. Ogden, Robt. C. Cornell, Bache McE. Whitlock, Geo. C. Cobbe, S. A. Reed, Prof. D. G. Eaton, Dr. D. H. Cochrane, Geo. S. Schofield, Hon. Stewart L. Woodford, William Jay, John McKean, Prof. H. C. Bartlett, Denniston Wood, Prof. A. J. Du Bois, F. L. Gilbert, J. B. Taylor Hatfield, Foster C. Griffith, Hon. Thomas C. E. Ecclesine, Malcolm Campbell, Lefferts Strebeigh, R. H. Buehrle, Prof. H. A. Newton, Wm. Hillhouse, M.D., Jas. L. Onderdonk, Wm. B. Patterson, Prof. J. E. Kershner, Francis M. Eagle, Warren Bigler, C. J. H. Woodbury, Rev. Peter J. Desmedt, D. H. Harsha, Prof. J. W. Nicholson, Prof. Peter S. Michie, Lieut. S. W. Roessler, E. F. MiUiken, Wm. P. Humbert, Chas. E. Emery, Prof. A. B. Nelson, Adam McClelland, D.D., Prof. H. T. Eddy, Miss H. L. Baquet, and others not wishing their names published, I copy each. Warned by the experience of others, the author was con- PREFACE. •. Vll vinced from the first that he could hope to derive no pecu- niary profit from a work like the present. But if it is now possible that there may accrue to him some snlall financial return, this possibility is due to the liberality of his publishers, who, although consulted late, and knowing the unremuner- ative character of the work offered, proposed voluntarily to undertake its publication upon terms more favorable than those which he had been endeavoring to secure. The acknowledgments of the author are due also to his printer, S. W. Green's Son, for the general excellency of the proof furnished, and especially for his uniform readiness to do, without regard to trouble, whatever was indicated as tending to render the work more correct in minor points. But while the author has, in the particulars mentioned, received much assistance from friends, to whom he would return his unfeigned thanks, he deems it but just to himself to say that he has never enjoyed the acquaintance of any one who had made the Calculus of Variations the subject of extensive study, and has consequently been obliged to depend solely upon his own judgment and the various works which he has consulted. It is not therefore believed that the present treatise can be entirely free from mathematical errors ; and hence the author would respectfully ask his readers, and especially those among them who may have given previous attention to this subject, to indicate any points in which his methods or results appear erroneous, or any places in which misprints may have been allowed to pass unnoticed. L. B. CARLL. Flushing, Queens Co., N, Y., July 8, 1881. CONTENTS CHAPTER I. MAXIMA AND MINIMA OF SINGLE INTEGRALS INVOLVING ONE DEPENDENT VARIABLE. Section I. Case in which the Limiting Values of x, y,y\ etc., are given. Prob. I. To find the shortest plane curve between two fixed points ; Mode of comparing curves, page i. Meaning of the term variation, and method of de- noting it, 2. No variation of the independent variable necessary ; Curve varied by means of its tangents only, 3. Development of 81, or the difference of length of two consecutive curves, 4. Review of the theory of maxima and minima in the differential calculus ; A maximum not always the greatest, nor a minimum the least, value of a function, 5, Extension of this theory to present problem, 7. Faulty solution obtained, 8. 6>(«) = -~, 9. True solution obtained, 10. Why first method of solution failed, 11. Term of the second order examined ; Coeffi- cient of dy"^dx must be of invariable sign, 12. No maximum nor minimum when it vanishes permanently; Previous solution not absolute, 13. Power of the new method; The constants a and b determined, 14. Prob. II. To find the plane brachistochrone, or curve of quickest descent of a particle, between two fixed points, 15. Simplification by omitting constant factor; General solution, 16. Particular solution, i8„ Term of the second order positive; Limitation of solu- tion, ig. Occurrence of infinite quantities, 20. Case 2. Same problem with hori- zontal as independent variable, 21. Ordinary maxima or minima of functions involving several variables, 22. Extension to present problem, 23. dy and dy not independent, 24. Terms of the second order examined, 26. Limitation of the solution, 28. Derived curve and primitive curve, 29. Prob. III. To find the curve which, with its evolute and extreme radii of curvature, shall enclose a min- imum area, 30. Terms of the second order, 33. Limitation of solution, 35. X CONTENTS. Section II. Case in which the Litniting Values of x only are given. General discussion, 35. Mode of transforming dUhy integration, 37. Final form of dU, 39. The coefficients of (5>o, <5>i, 8ya' , 8yi , etc., must severally van- ish, 40. The coefficient of dydx in the integral must also vanish, 41. Prob. IV. To maximize or minimize the expression / f {y") dx, 42. Prob. V. To nx-i limize / (/'^ — 2y) dx, 44. Probs. I., II. and III. re- 'Xo maximize or minimize 'x^ sumed, 48. Number and determination of the constants in the complete integral oi M := o, 52. Three exceptional cases, 55. Integrability of M=o; Formulae (A), (B), (C) and (D), 61. Prob. VI. To find the solid of revolution which will experience a minimum resistance in passing through a fluid in the direction of the axis of revolution, 66. Prob. VII. To find the surface of revolution of mini- mum area, 70. Prob. VIII. To find formula for terms of second order when pxi U-= I yf{y')dx, 77. Application to Prob. VI., 80. VfXo Section III. Case in which the Limiting Values of x also are variable. Prob. IX. To find the curve of minimum length between two curves, all in the same plane; Mode of varying the required curve when x^ and xi are variable, 81. General solution must be independent of conditions at the limits, 84. 8y^, dx^, 6yi and dxi cannot be independent, 85. Formulae for eliminating dyo and dvi, 86. Application to present problem, 87. Terms of the second order considered, 88. Prob. X. To find the plane brachistochrone, the particle starting from any hori- zontal, 90. General theory of the subject, 93. Prob. XI. The evolute problem with limiting curves, 94. Prob. XII. Minimum surface of revolution with varia- ble limits, 95. Section IV. Case in which some of the Limiting Values of x, y, y', etc., enter the General Form of V. Prob. XIII. To find the brachistochrone between two curves, the particle start- ing from the upper curve, 96. Terms of the second order, loi. General theory of the subject, 104. CONTENTS. XI Section V. Case in which U is a Mixed Expression; that is, contains an Integral together with Terms free from the Integral Sign. nx-^ y" Prob. XIV. To maximize or minimize / y") — dx, 105. General theory of the subject, 107. ^"^^ ^ Section VI. ) Relative Maxima and Minima. Prob. XV. To find the plane curve of given length which, with its extreme ordinates and the axis of x, shall contain a maximum area; Restriction of derived curves and use of the terms absolute, relative and isoperimetrical, 108. Why- ordinary method of solution fails, 109. Special condition established, no. Solu- tion effected, in. Determination of the constants, 113. Bertrand's proof of Euler's method, 114. Immaterial whether we regard the curve or area as con- stant, 116. Terms of the second order considered, 117. Problem considered when limits are variable, iig. Prob. XVI. To find the solid of revolution of given surface and maximum volume, the generatrix meeting the axis of revolu- tion, at two fixed points, 122. Constants determined; First notice of discontinuity, 124. Condition when limits are variable, 126. Character of the generatrix when unrestricted, 131. Prob. XVII. To find the curve which a uniform cord must assume when its centre of gravity is as low as possible, 133. Prob. XVIII. Case of James Bernoulli's problem, 138. Prob. XIX. To find the solid of revo- lution of given mass whose attraction upon a particle situated upon the axis of revolution shall be a maximum, 141. Prob. XX. To find, as in Prob. VI., the solid of minimum resistance, supposing its mass and base to be given, 144. Prob. XXI. To find the solid of revolution of given mass and minimum moment of inertia with respect to an axis perpendicular to that of revolution through its middle point, 148. Section VII. Case in which V is a Function of Polar Co-ordinates and their Differential Coefficients. Prob. XXII. To find the path of minimum action for a revolving particle, attracted toward a fixed point, according to the Newtonian law, 151. Prob. XXIII. To find the plane closed curve of given length which will enclose a maxi- mum area, 158. XU • CONTENTS. Section VIII. . Discri7nmation of Maxima and Minima {Jacobi's Theorem). Case I. V supposed to be a function of x, y and y' only, 163. Rules for apply- ing theorem in this case, 174. Prob. XXIV. To apply theorem of Prob. I., 175. Prob. XXV. Application to the first case of Prob. II., 176. Prob. XXVI. Application to Prob, XXII., involving polar co-ordinates, 178. Geometrical illus- tration of theorem for this case, 182. Prob. XXVII. To apply theorem to I y^f{y)dx, 184, Discussion of Prob. VIII. completed, 188. Prob. XXVIII. To apply theorem to Prob. VII., 189. Probs. XXIX. and XXX. To apply theorem to the second case of Prob. II., and also to Prob. XVI., 193. General discrimination in cases of relative maxima and minima ; Euler's method and Jacobi's theorem defective, 197. Two exceptions to Case i, 202. General view of Jacobi's method, 204. Two lemmas, 206. Case 2. V supposed to be a function of x, y, y' and y" only, 20g. Prob. XXXI. To apply theorem to Prob. v., 226. General rules for applying theorem in this case, 230. Three excep- tions to Case 2, 231. Case 3. V supposed to be a function of x,y,y', .... I'W, 232, Four exceptional forms of F, 241. Connection of the variations of U when we take successively x and y as the independent variable, 242. Section IX. Discontinuous Solutions. Prob. XXXII. Maximum solid of revolution resumed, 248. Failure of former methods, and discontinuous solution proposed, 250. Sign of dy, 8y\ etc., not always in our power; M not always zero; No appeal to terms of second order necessary, 251. Discontinuous solution confirmed by foregoing principles, 253. Cause of discontinuity; Explicit and implicit conditions, 255. Prob. XXXIII. -Prob. V. resumed with conditions, 256. General theory of discontin- uous solutions, 264. Prob, XXXIV. To solve Prob. VII. when a catenary is impossible; General discussion, 267. Two solutions admissible, and formulae for determining which will give the smaller surface, 273. ProTa. XXXV. To determine the path of least or minimum action for a projectile acted upon by gravity only, 275. Jacobi's theorem applied to problem, 279. Discontinuous solutions considered, 281. Formulae when two solutions are admissible, 283. Prob. XXXVI. To seek a discontinuous solution for Prob. XXII, involving polar co-ordinates, 285. Prob. XXXVII. To determine under certain conditions the CONTENTS. Xlll quickest or brachistochrone course of a steamer between two ports on a stream, 287. Prob. XXXVIII. The brachistochrone course of a vessel when length of course is fixed, 291. Prob. XXXIX. To maximize or minimize / \ ay"-^-\-— l dx, 299. Prob. XL. To determine the discontinuous solution in Prob. XV., 302. Prob. XLI. To determine the discontinuous solution in Prob. XIX., 306. Prob. XLII. To find the discontinuous solution in Prob. XXL, 309. Prob. XLIII. To find the discontinuous solution in Prob. XX., 311. Prob. XLIV. To find the plane brachistochrone between two points, the particle not to pass with- out a certain circular arc, not exceeding a quadrant, 322. Section X. Other Methods of Variations. First Method. Variations ascribed to x also throughout U\ Method illustrated geometrically, 328. General expression for 8 U, 329. Expressions for dy' , 8y\ etc., 330. (5 i/ reduced and compared with the usual form, 331. General solu- tions, and also terms at the limits, the same as by previous method, 332. Merits of the two methods, 334. Second Method. The arc ^ taken as the inde- pendent variable, but not varied. Prob. XLV. To maximize or minimize / ^ f {x> y)Vi-\-y^dx; J made the independent variable, 335. Method illus- trated ; s must be varied at the limits, 336. 56^ obtained ; dx and dy not inde- pendent, 337. Method of Lagrange introduced, 338. Expressions for dx', dx", etc., dy', dy", etc., 339. Formulae for dxi and dyi when limits are fixed, 341. Same when limits are variable, 342. General formula of solution, 344. Applica- tions ; Final limiting equations the same as when x is the independent variable, 345. Prob. XLVI. To maximize or minimize / (v Vi -^ y'^ -{- n) dx = (v -|- Mx') ds, V and u being functions of x and y only, 346. General for- t/so 'So mula of solution, 348. Applications, 349. Prob. XLVI I. To maximize or minimize / Vds, V being a function of r, the radius of curvature of a plane t/so curve, 350. Application to Prob. III. and to the problem of the elastic spring, 353. Third Method. Vzx'idilions ascribed to the independent variable j-; Method illustrated, 354. General form oidU, 355. Expressions for dx',dx", dy' , dy", etc., 356. dx and dy not independent, and method of Lagrange necessary, 357. Method applied to Probs. XLV., XLVI. and XLVIL, 358. XIV CONTENTS, CHAPTER II. MAXIMA AND MINIMA OF SINGLE INTEGRALS INVOLVING TWO OR MORE DEPENDENT VARIABLES. Section I. Case in which the Variations are unconnected by any Equations. Prob. XLVIII. To find the shortest curve in space between two points, curves or surfaces; Former principles extended, 363. Expressions for 5y">, 52<»), <5w("), etc., 364. Constants determined, 366. Terms of the second order examined, 367. x^ and xx variable, 368. Prob. XLIX. To find the brachistochrone in space for a particle descending from one fixed point, curve or surface to another, 372. Number and determination of the constants in the general solution of problems of this class, 374. Prob. L. To establish the principle of least action, 375. Section II. Case in which the Variations are connected by Equations, Differential or other. Prob. LI, To find the shortest or geodesic line upon the surface of a sphere, 379. General discussion of the method of Lagrange, 384. Number of the con- stants and ancillary equations, 385. Lagrange's method extended, 386. Prob. LIL To maximize or minimize / v V 1 -\- y"^ -\- z'^ dx, v being a function of Jxq X, y and z, the connecting equation being /(.Jtr, y, z) =0 , 388. Prob. LIII. To find the shortest or geodesic curve upon any surface, 396. Prob. LIV. To find the geodesic curve upon the surface of a spheroid, 399. Prob. LV. Tp find the brachistochrone upon a given surface, the particle being urged by any system of forces, 401. Prob. LVI. To find the curve of minimum length be- tween two points in space, the radius of curvature being a constant, 405. Dis- continuous solution, 410. Miscellaneous observations, 418. CONTENTS. XV CHAPTER III. MAXIMA AND MINIMA OF MULTIPLE INTEGRALS. Section I. Case in which U is a Double Integral, the Limiting Values of x, y, z, etc. , being fixed. Prob. LVII. To find the surface of minimum area terminating in all directions in a given closed linear boundary; Mode of comparing surfaces, 422. View of double integration, 423. Projected contour, and sign of substitution, 424. Varia- tions introduced, 427. Principles of maxima and minima extended; Formulae for 8p, dq, dr, 8t, etc., 429. Solution effected, 430. Terms of the second order; Limitation of solution, 434. Form of M and nature of its integral when F is a function of x, y, z, p and q only, 435. Prob. LVIII. The intersept problem, 437. Section II. Forjnulce necessary for the Transforinatio7i of the Variations of a Mutiple Ifitegral. 439 , Section III. Maxima and Minima of Double Integrals with Variable Limits Prob. LIX. To discuss Probs. LVII. and LVIII. when the limiting values of x and y only are fixed ; New form of 8 U, and terms at the limits explained, 447. Conditions when ^is to be a maximum or minimum, 450. Results applied to problem, 452. Prob. LX. To discuss Probs. LVII. and LVIII. when the required surface is to touch one or m.ore given surfaces; that is, when the limit- ing values of x and y are also variable ; x and y varied at the limits only, 454. Mode of obtaining 5 f/ to the second order, 455. Terms of the first order only con- sidered, 459. Some limiting surface necessary for a maximum or a minimum; Equations between dz, Dy and Dx at the limits, 461. Final form of terms at the limits, 463. Application to present problem, 464. Mode of determin- ing the arbitrary functions involved in the complete integral of M =0, 465. XVI CONTENTS. Occurrence of infinite quantities, 469. Prob. LXI. To maximize or minimize / / ^p'^-\-(f dy dx, 470. Prob. LXII. To maximize or minimize I ^ I ^ {z — px — qy)dy dx, while I I Vf^-[-q^dydx remains con- stant; Mode of extending Euler's method, 475. Terms of the second order, 480. Prob. LXI 1 1. To find the surface of lowest centre of gravity, 481. Prob. LXIV. To find the minimum surface covering a given volume upon a plane, 483. Prob. LXV. To discuss the case when K is a function of x, y, z, p, q, r, s and /, 485. Imperfection of preceding theory, 490. Exceptional forms of V, 492. Section IV. Extension of Jacobi's Theorem to the Discrzinination of Maxima and Minima of Double Integrals. 493 Section V. Maxima and Minima of Triple Integrals. Prob. LXVI. To minimize j-^^ H' H^ ^i +/-' +q^-{- ^dzdydx, u being e/xo e/2/0 «^^o the density of a body of given form, position and mass ; Triple integration and limiting faces explained, 498. Mode of finding ^ U when the limits of x, y and z are fixed, 500. Conditions when ^ is to be a maximum or a minimum, 503. Problem solved, 505. Form of (J f/ when the limits of ;r, _j' and z are variable, 506. Conditions when U is to be a maximum or a minimum, 509. du, Dz, Dy and Z>.r not independent ; Equations between them, 510. Conditions resumed, 5". Section VI. Another View of Variations. General explanation, 514. Prob. LXVI I. To find the solid of maximum volume, 524. CONTENTS, XVU CHAPTER IV. APPLICATION OF THE CALCULUS OF VARIATIONS TO DETER- MINING THE CONDITIONS WHICH WILL RENDER A FUNCTION INTEGRABLE ONE OR MORE TIMES. Section I. Case in which there is but One Independent Variable. Prob. LXVIII. To maximize or minimize / -< " =V- + ^^ >■ dx, 533. M found to vanish identically, and U to be integrable ; General theory deduced for similar cases, 534. Prob. LXIX. To determine the conditions which will XQndQT f {x, y, y' , . . . . y^)) integrable more than once, 536. Prob. LXX. To determine the conditions which will render /(x,j,y, . . . . }^'^^, z, z' , . . . . s('»))^x immediately integrable, 537. Section II. Case in which there are Two Independe^it Variables. Prob. LXXI. To determine the conditions which will render / I f {x, y, z, p, q) dydx reducible to a single integral, 538. CHAPTER V. HISTORICAL SKETCH OF THE RISE AND PROGRESS OF THE CALCULUS OF VARIATIONS. 54I NOTES. Note to Lemma I., 557. Note to Lemma II., 560. Note to Art. 369, 562. Note to Art. 372, 568. CALCULUS OF VARIATIONS. CHAPTER I MAXIMA AND MINIMA OF SINGLE INTEGRALS INVOLVING ONE DEPENDENT VARIABLE. Section I. CASE IN WHICH THE LIMITING VALUES OF X, F, F', ETC., ARE GIVEN Problem I. I. Suppose it were required to find the shortest plane curve or line which can be drawn between tzvo fixed points. Let A CB be the required line, which is of course straight, and AEB any other line derived from the first by giving X ox indefinitely small increments to any or all of its ordinates, while the corresponding values of x remain unaltered. Then the line ACE must be shorter than the line AEB. This remark would be equally true if the changes in the 2 CALCULUS OF VARIATIONS. ordinates oi AB had not been made indefinitely small ; but then, even if the second line were shown to be longer than the first, we could not be certain that some third line, lying a little nearer the first, might not be shorter than either. Thus it will be seen that questions may arise which require an investi- gation of that increment which a curve would receive, not from any change in the values of x, nor in the values of the co-ordinates of the fixed extremities, but from indefinitely small changes in the values of y throughout the whole or a portion of the curve ; thus altering in a slight degree the functional relation which previously subsisted between x and y. 2. Now the general expression for the length of any plane curve between two fixed points is ^=/V^p"+^^ . (I) in which the suffix i relates to the upper, and o to the lower limit of integration, and this expression cannot be integrated so long as y is an unknown function of x. Hence, in determining the increment which will result to a curve from an indefinitely small change in its form, we shall be concerned with two species of small quantities : first, those changes which x and y undergo as we pass from one point to another indefinitely near or adjacent on the same curve, which are denoted by dx and dy, these being necessary for the general expression of / in (i) ; and secondly, that change which y undergoes as we pass from a point on one curve to a point on another curve indefinitely near or adjacent, the value of X being unaltered. These latter quantities are called variations, and are denoted by the Greek letter d, delta, or d. Thus dy is read, the variation of y ; -j— , the variation of -7-, etc. As another illustration of the difference between these two SHORTEST PLANE CURVE BETWEEN TWO POINTS. 3 classes of quantities, we might say that dy as used in (i) is the difference between two consecutive states of the same function of x, while Sy is the difference between two consecu- tive or adjacent functions taken for the same value of x. The use of this symbol S is due to Lagrange, and while it prevents confusion, it also suggests the character of the variation as a species of differential. It is plain that we can vary the form of a curve which terminates in two fixed points in any man- ner we please, by simply giving suitable changes to its ordi- nates without varying its abscissas, and we shall therefore at present ascribe no variation to the independent variable x, but simply to the dependent y or to its differential coefficients wnth respect to x, 3. Resuming equation (i), we will now show how to find SI, or that increment which / would receive, not from any change in the limits of integration, but from an inappreciably small alteration in the value of jr as a function of x. We. shall dy d^v in general put y' for -p, y" for ^i-^, etc. Then we have dx"" -k dv" dx^ + df = — ^F^ d^ = (I +y^) dx^; hence (i) becomes ^=ff'y^i~+rdx. (2) 1 It will be seen that y does not occur directly or explicitly in the last equation ; but since y' represents the natural tan- gent of the angle which a tangent to the curve at any point makes with the axis of x, it is clear that the form of this curve can be also altered at pleasure by giving suitable variations to the slopes of these tangents, and that if these variations be 4 CALCULUS OF VARIATLONS. indefinitely small, the remarks that have been made regarding dy will be equally true regarding dy' . Equation (2) may be written where Now in F change y' into y + Sy'. Then the new state of V, being denoted by V\ may be developed by the extension of Taylor's Theorem, thus : where, following the analogy of differentials, we write 6/% d/\ etc., for {dyj, (dyj, etc. Hence, if we call V- V^SV, we have in w^hich -^„ etc., are the partial differential coefficients of V with respect to y . whence, if we change V into V\ dx remaining unaltered, and denote the new state of / by /', we shall have V = V'dx, and calling V — /, dl, we arrive at the equation SHORTEST PLANE CURVE BETWEEN TWO POINTS. 5 4. Before proceeding it may be well to advert to the theory of maxima and minima, as developed by the differ- ential calculus- A function is said to be a 'maximum when its value is greater, and a minimum when its value is less, than that which it would have if any or all of its variables should receive indefinitely small increments, either positive or negative. Thus while the greatest value of a function, if not infinite, is always a maximum, it does not follow that every maximum is the greatest value of which the function is capable. Neither is the greatest value in every case the only maximum. The foregoing remarks apply equally to a minimum, it being only necessary in either case to compare the supposed maximum or minimum state of the function with the value of the states which immediately precede and succeed it. Taking, for simplicity, a function of a single variable, this state is determined and comparison effected as follows : Let / be any function of x and constants, and change x into x -J- //. Then if we develop f , the new state of the function, by Tay- lor's Theorem, and subtract the original state, we shall have f'-f=t'^^h:^'''-^-^ w h being either positive or negative. We shall denote this series by 5. Then, if / is to be a maximum or minimum,/'— /must be negative in the former case and positive in the latter, independently of the sign of h. But il no differential coefficient in S become infinite, and we make // indefinitely small, the sign of S will either depend upon that of its first term, which cannot be independent of h. O CALCULUS OF VARLATIONS, or, if that term reduce to zero, upon the sign of the first that does not. Now if this term be of an odd order, its sign would be affected by any change in that of h ; but if of an even order it would not, since h must be real. Hence any value of x which would render /a maximum or minimum must at least satisfy the equation -3— = o, and the roots of this equation furnish us with trial values of x^ which, when substituted in the remain- ing terms of 5, must render the second term negative for a maximum and positive for a minimum, or must fulfil the same condition for some other term of an even order, having reduced those which preceded it to zero ; and we must reject those values of x which do not satisfy these conditions. It may also be useful to observe that -j- does not repre- sent the exact ratio of the increments of / and x, dx being infinitesimal, but merely the limit of that ratio ; that is, the value toward which it may be made to approach to within any assignable limit, but which it can never actually equal, it being meaningless to say that dx ever really becomes zero. Or, better, we may regard -j- as merely a function derived from f by certain algebraic methods which accord with the rules of differentiation ; and the same remarks will apply to the higher differential coefficients of /. Hence, since these coefficients are entirely Independent of any increment which / actually receives, we may, without altering any of them, replace h in (4) by dx^ Sx^ or any other infinitesimal we please. 6. If the roots of the equation -j- — o comprised all the values of x which could render / a maximum or minimum, still, since / might be capable of several maxima or minima, / SHORTEST PLANE CURVE BETWEEN TWO POINTS. J we would have to determine which maximum would be the greatest, or which minimum the least; although the deter- mination would in general be easy enough. But the equa- tion in question does not give all the required values of x. For, if any of the differential coefficients in (4) become infinite, the reasoning of the last article will no longer hold true. In fact, it is well known that / can become a maximum or mini- mum when its first differential coefficient is infinite, or when the same is finite while the second is infinite. These instances are examples of what are often termed failing cases of Tay- lor's Theorem — although, strictly speaking, the theorem does not fail at all, only the development becomes useless from its indeterminate character, and that not from any imperfection in the theorem itself, but owing to the existence of such con- ditions as to render impossible an entirely finite development of the form required. 6. Since the' value of h in (4) is altogether independent of its coefficients, and might be replaced by dx, dx, or any other symbol we please, it is clear that the form in which we have expressed d/ in (3) is analogous to that of 5 or/'—/, except that each term in SI is multiplied by dx, and is under an integral sign, and that the function taken is one of y and con- stants, among which x is reckoned. Considering the first term of that expression, viz., / dy we see that by taking Sy indefinitely small throughout the curve we may ultimately render this term greater than the sum of the others, unless, indeed, that integral becomes zero for all possible values of Sy; it being understood that the variation of any quantity is to be always infinitesimal as com- pared with that quantity. It is also clear that if we change the sign of Sy throughout the integral — that is, of each sy. ■o CALCULUS OF VARIATIONS. leaving its minute numerical value unaltered — we shall also change the sign of the above integral, while the sign of the succeeding integral in (3) will remain unchangedi 7. From an examination of the figures, Art. i, it will be seen that if ACB be the minimum line between two fixed points, and we draw a second in any manner we please by giving infinitesimal variations to y' , we may also draw a third line by giving to y' variations numerically equal but of oppo- site sign. Then, ^\\\cq ACB is a minimum, I' — I or 61 must be positive ; /' being the length of either of the lines ACB, Hence, from the reasoning of the last article, we must have since otherwise 61 could not be of invariable sign, as its sign would be the same as that of the above integral, which could be made to vary by changing that of 6y' . Moreover, the sec- ond term in 6L viz., 11 rr must become positive; or if it reduce to zero, some other term of an even order must become positive for all values of dy\ all the preceding terms having reduced to zero. But, as in the differential calculus, the foregoing is based upon the supposition that none of the differential coefficients of V in (3) become infinite within the limits of integration, or, in other words, that V —V is throughout these limits capa- ble of a finite development by Taylor's Theorem, where V denotes what V becomes when we change y' into y'-\- 6y' , 8. We may now proceed to a full solution' of the problem. We have SHORTEST PLANE CURVE BETWEEN TWO POINTS. V=V,+y\ ^y- ^-^^ y. d'V I _ I dW _ 3/ _ 3/ dy- - ^(i+/T ^" "^y^ ^i+/T v^ Hence, as these and the succeeding partial differential coeffi- cients of Fwith respect to y' are all finite, we can develop /' by Taylor's Theorem, and equation (3) gives in which we have first to consider the expression £/-sydx = o. (6) v' This equation is of course satisfied by making ~ zero, which gives necessarily y' zero, and y a constant. This would make the required curve a right line, coinciding with, or par- allel to, the axis of x. While this solution is correct so far as the general form of the required curve is concerned, it will not be always possible to draw such a line through two fixed points given at pleasure, unless we are at liberty to assume the axis of x so as to make y^ and y^ equal, which is not con- templated. We must, therefore, seek another solution. 9. We will begin by transforming ^ y' thus: / dy ^ dx Change y into y + Sy^ while ;r, and consequently dx, undergo TO CALCULUS OF VARIATIONS. no alteration. Then denoting the new value of y' by Y' , we have Whence, subtracting from the first member y\ and from the last its equal -^, we have dx V -y-- dSy ~ dx' t y -y= ■h'. whence 6/= ddy dx' like manner, /= dy dx'' Change y into y-^-^y- Then dx'^-^^ -^^ dx'^ dx'' Y"—y" •^ dx' ' and, similarly, -^ dx"" where n is any positive integer. 10. Equation (6) may now be written ^x^ V dx But integrating by parts, we have SHORTEST PLANE CURVE BETWEEN TWO POINTS. II where the suffix i denotes what the quantities affected become when X is x^, and o what the same quantities become when x is x^. But since the two points through which the required une must pass are fixed, Sy^ and dy^ are each zero ; that is, y receives no increment at these points, and therefore (8) becomes This equation can be satisfied by writing d y' y Squaring, clearing fractions, and transposing, we have y- _ cy^ = c\ /=: -M=. = a, y = ax + d, Vi — ^ the general equation of the straight line. 11. It will be seen that the solution y=o is only a par- ticular case of the more general one just obtained, and we are therefore led to inquire why the method pursued in Art. 8 did not give a satisfactory result. Now, since we have the equations Sy' = — -^, ^- = Cy (6) may be written ax V £?V tf/ dx = £\dSy = o, whence, by integration. 12 CALCULUS OF VARIATIONS. and because both dy^ and 8y^ are zero, this equation can be satisfied without making c zero. The error, therefore, in Art. 8 appears to have arisen from the fact that we required the curve to pass through two fixed points, anii then entirely disregarded that condition in obtain- ing our solution. But (9) was estabhshed by expressly impos- ing this condition upon the problem; and as there are no further conditions to be imposed, and as ^y cannot be further transformed, that equation can only be satisfied by equating to zero the coefficient of dy dx in that equation. 12. Resuming equation (5), let us next consider the term of the second order, Jxa ''~Sy-dx. (10) If the solution given above be a true minimum, this term must become positiv^e, or must reduce to zero. Now since X is the independent variable, dx is always supposed to be estimated positively ; and as d/^ can never be negative, if we also regard Fas positive, we see that every element of (10) is positive, and that consequently the integral itself must be of the same sign. We conclude, therefore, that a right line is the plain curve of minimum length between two fixed points. If the coefficient of Sy'^dx in (10), which we may call Z, could have changed its sign within the given limits of inte- gration — that is, if Z could have been positive throughout some portions of the curve, and negative throughout others — we could make (10) take either sign, and there could be neither a maximum nor a minimum. For by var3^ing y' throughout those portions of the curve for which Z was negative, while leaving the other portions unvaried, the inte- gral would become negative, or by pursuing an opposite SHORTEST PLANE CURVE BETWEEN TWO POINTS. 1 3 course it would become positive. Hence, in this and similar cases, the coefficient of dy''^dx must be of invariable sign for all values of x from x^ to ;r,. If Z could have reduced to zero throughout the Avhole range of integration, thus rendering the integral itself zero, we might generally infer that the solution was neither a maxi- mum nor a minimum. For in order to the existence of either, the term of the third order involving Sy'^ must also vanish, which would seldom if ever occur. It will be observed that the term of the second order is positive whether the extremities of the required curve are supposed to be fixed or not. But if we disregard this con- dition, the terms of the first order would not vanish, so that we would not obtain a minimum, except, indeed, we adopt the particular solution of Art. 8. We shall, however, subse- quently show that when the limiting values of x only are given — that is, when the required curve is merely to have its extremities upon two fixed lines perpendicular to the axis of X — the solution of Art. 8 is that which must be taken. 13. In the preceding discussion we have merely proved that the straight line between two fixed points is shorter than any other plane curve which could be derived from it by making indefinitely small changes in the inclination of its tan- gents to the axis of x, either in certain portions or through- out its whole extent. We could not, therefore, by the use of the calculus of variations alone, become certain that the straight line is the shortest plane curve which can be drawn between two fixed* points, but merely that it is a curve of minimum length, the existence of other minima being possible ; one of which might, perhaps, be less than the present, and might itself be the shortest curve. Again, the precpding method does not permit us to com- pare the straight line with all other plane curves which can be drawm indefinitely close to it. For in developing /', Art. 3, 14 CALCULUS OF VARIATIONS. we were obliged to ascribe indefinitely small increments or variations to y' only, since y did not directly or explicitly occur in /. Hence the curve which we derive by variations can have no abrupt change of direction ; because no such- change could occur without rendering dy' appreciably large at that point. Therefore all curves with cusps, and all systems of broken lines, are excluded from the comparison, although it is evident from the figure that such curves might be drawn without making the variations of y appreciable, but only those of/. (4. From the remarks of the preceding article, which were deemed 'necessary in order to guard the reader against certain misconceptions which are common among students of this subject, it must not be inferred that the calculus of vari- ations is of little use as a method of solving questions of max- ima and minima. For we shall see as we advance that it can in general be made to give a satisfactory solution when such a solution exists. Indeed, the recent discoveries relative to the theory of discontinuity, which are due chiefly to the labors of Prof. Todhunter, and of which we shall speak hereafter, show that this branch of the calculus does not in reality fail to present solutions even in very many of 1;fiose cases in which its failure has been hitherto assumed. !6. It remains only to determine the constants a and b which occur in the general solution. It will appear that since the required line is to pass through two fixed points whose co-ordinates are x^, y^, x^, jj/„ we must have BRACHISTOCHRONE BETWEEN TWO POINTS. 1 5 and therefore so soon as these quantities are given a becomes known. Then to determine b, we have jKo= <^^A- ^^ b = y — ax,=y — ^^^^'^ ,r„, and thus b is also known when x^, x^, /„, j\, are fully given. 16. In further illustration of our subject we next proceed to consider another problem, the solution of which is not so generally known. Problem II. It is required to determine the equation of the plane curve, doiun which a particle, acted upon by gravity alone ^ would descend from one fixed point to another in the shortest possible time. Let a be the upper and b the lower point. Assume the axis of x vertically downward, and a as the origin of co-ordi- nates. Also let the variable s be the length of the required curve at any point measured from a ; v, the velocity of the particle at the same point ; and /, its time of descent from a to that point. Then we wish to determine the curve which will render T a minimum, where T is the total time of descent from a to b, or what t becomes at the point b. We must first then find / as a function of x and j/, or their differentials. Now, from the well-known differential equations of motion in mechanics, we have dt^'^' (I) v and, Art. 2, ds = Vdx' + d/= Vi +/' dx. 1 6 CALCULUS OF VARLATIONS. We also know that the particle loses no velocity in pass- ing from one point to another of a curve with no abrupt change of direction, and that therefore, if it start from a state of rest at a, its velocity at any point of the curve must equal that which it would have acquired in falling freely through the same vertical distance. Hence we shall have V = V2gXy g being the acceleration due to gravity. Therefore (i) becomes y2gx and T= / , ~ dx, (2) which is to become a minimum. 17. But since ^ is a constant, the second member of (2) may be written ^' ^' +^" d.. V2g ^^^ Vx Now, it is evident in general that if c times any integral is to be a maximum or a minimum (c being any constant), the in- tegral itself must also be a maximum or minimum. Hence, omitting the constant factor, the expression to be rendered a minimum in this problem may be written U= f — -"t^" dx = n Vdx, (3) Now, as in* the preceding problem, change y 'mX^o y' -\-^y' and develop by Taylor's Theorem. Then we shall obtain BRACHISTOCH-RONE BETWEEN TWO POINTS. 1 7 = / ——J=Sy'dx^ / :=. • dyV4r + etc. (4) We shall not in future develop any variation beyond the terms of the second order, since if the terms of the first two orders should become zero, there could rarely if ever be either a maximum or a minimum, as explained in Art. 12. Hence we must have r^ —=jL=Sy'dx = o, (5) But since the two extreme points are fixed, we must impose this condition upon the problem by integrating (5) by parts, as in the preceding problem, and neglecting the terms thus freed from the integral sign, because containing 6y^ and dy^. Performing this operation, we shall obtain - r'~--=jL= dydx^o, (6) = 0, (7) = c. (8) dx \/^{^i _|.y2) d y y Vx{i +y^) Now since c in the last equation is an arbitrary constant, make 1 8 CALCULUS OF VARIATIONS. it equal to -=. Then squaring, clearing fractions, and trans- ^ a ■ . posing, we have '-— 'J^ — £ a a Whence solving fory, we obtain y = . > (9) Va — X which is known to be the differential equation of the cycloid. Therefore J/ = versin -^^ x ~ Vax — x^-\- b, (lo) x where a is twice the radius of the generating circle, and b is zero, because the origin was taken at the upper point. The last equation may be finally written thus : j = r versin"^ \^2rx — x'', (ii) where the circular function is natural, and r is the radius of the generating circle. 18. By disregarding the condition that the curve must pass through the two fixed points, we shall, as in the preced- ing problem, obtain from (5), y' z=:0, VxiT^y'^) which makes y' zero, and y a constant, which must also be zero, because the curve passes through the origin. There- fore the curve would in this case coincide throughout with^ BRACHISTOCHRONE BETWEEN TWO POINTS. 1 9 the axis of x, which solution could only be possible when the two points were in the same vertical line, and then its truth is self-evideht. 19. Let us now consider the term of the second order, viz.. Jxa ^sy^dx, (12) ^0 2|/;r(l+y7 If the cycloid be the true solution of our problem, this term must become positive, whether y' be varied throughout the whole integral or only throughout certain portions taken at pleasure. To satisfy this condition it is merely necessary that Z, the coefficient of Sy'dx in (12), shall become positive and not change its sign as we pass from a to b. But since x can- not become negative in this problem, the square root of x is real and may be considered as always positive from a to b \ then, as we may regard Vi -\- y'"^ as always positive, the above conditions are satisfied, and we conclude that the cycloid, having a cusp at a, its base horizontal, and its vertex down- ward, is a solution of our problem. Let us also try the solution / = o of Art. 18 ; this will reduce (12) to J. •"• 2Vx which will also become necessarily positive if we assume \G: to be positive. Thus this solution likewise, when it is pos- sible, renders T a minimum, as it evidently should. 20. Remarks similar to those made in Art. 13 apply also to this example. For it is plain that we have only compared the cycloid as a curve of descent, with all other curves pass- ing through the given points, having no abrupt change of direction, and drawn indefinitely near to it. Hence we have 20 CALCULUS OF VARIATIONS. in reality only shown that the cycloid is one of the curves which renders T a minimum, the term minimum being used in the technical sense hitherto explained. However, as in the fornier problem, these restrictions are merely theoretical, and are noticed in order to prevent misconceptions which might occasion difficulty in subsequent discussions. For in the present case the cycloid between two points is undoubtedly the curve of quickest descent from one to the other, and from this property it is often called the brachisto- chrone. 21. In addition to what has been already said, we must here call attention to another point which is often passed over by elementary writers on this subject. Suppose y' to become infinite for some point within the range of integra- tion, as it does at the vertex of the cycloid. Then when we change y intoy -\- Sy' , if we regard, as we must by the theory of the subject, Sy' as taken arbitrarily, but always indefinitely small, we can make the new or derived curve assume any form we please, except that its tangent at X must have the same direction as that of the cycloid at the vertex, where X IS the abscissa of the vertex. For suppose the vertex tangent of the cycloid to undergo a slight change of direction, so that its new angle of inclination to ;i' may differ from a right angle in an indefinitely small degree. Then we cannot assert that this small change of direction could be produced by an in- definitely small change in the value of y\ or the natural tan- gent of the right angle. That is, owing to the indeterminate nature of infinity, we cannot with certainty apply the method of variations to any element of the integral which is affected by an infinite value of y' , and hence the integral must not be extended so as to include this element. In the present case, then, we are only sure of a minimum so long as we are not obliged to go beyond the vertex of the cycloid for b. But the occurrence of an infinite value of y' in any case PLANE BRACHISTOCHRONE- BETWEEN TWO POINTS. 21 will not warrant us in concluding that the solution does not give a true maximum or minimum, even when the integral includes that value of y'. All that we can say is that the pro- posed method becomes inapplicable. Indeed, we shall have occasion to show that sometimes, by changing to polar co- ordinates, or by some other change of the independent vari- able, the integral may in these cases be freed from infinite quantities, and the previous solution shown to give a true maximum or minimum. Of course if we regard Sy' as zero when y' becomes in- finite — that is, consider the tangent to the curve as fixed at that point — the variation of the element becoming zero, may be included in the development, and all difficulty disappears. It will be observed that F becomes infinite at A, and the solution is therefore still subject to any objection, but there would seem to be none, which can arise from this fact. Case 2. 22. As a means of still further extending our knowledge of variations, let us resume the preceding problem, merely taking the horizontal as the axis of x. Then, the notation and the other conditions being un- changed, we must, as before, render T a minimum. But, as formerly, ds dt = -^, ds = Vdx'+d/ = Vi +y'dx, V where y' now means the natural tangent of the angle which any tangent to the curve makes with the horizontal instead of the vertical axis. Also, v = V2gy, so that, neglecting, as be- before, the constant factor, we must minimize the expression 22 CALCULUS OF VARIATIONS. Now in V change y into y + dj, and / into / + Sy' . Then we may develop V , or the new state of F, by the extension of Taylor's Theorem, thus : We also have u' ^ r'v'dx, where U' is what ^becomes when we change Finto V-\-dV or into V, dx being unaltered. Hence caUing U' — U, dU, we have Indeed, it is evident that a similar course could be pursued should V contain any number of quantities capable of being varied. 23. It may be well before proceeding further to refer briefly to the subject of maxima and minima of functions involving more than one variable, as it is developed by the differential calculus. Let / be a function of x, y, z, etc. Give small increments, k, i, k, etc., to X, y, z, respectively, and develop /', the new state of /, by Taylor's Theorem. Then the terms of the first order in /'—/will be ^^ I ^/;4_^/ ^j_etc PLANE BRACHISTOCHRONE BETWEEN TWO POINTS. 23 which must collectively vanish ; and if the quantities h, i, k, etc., be independent, each of the partial differential coeffi- cients of / must also vanish. Then the terms of the second order. 2 \dx^ dxdy ' dy^ must become collectively negative for a maximum and posi- tive for a minimum. Also, if the increments be independent, the second partial differential coefficients of / must fulfil cer- tain conditions among themselves, for an account of which, as they have no application here, the reader is referred to works on the differential calculus. 24-. The expression for SUm. (i) is similar to that for/'—/, only each term is multiplied by dx, and is under an integral sign, Sy and Sy' taking the place of h and i, dx being regarded as constant. In the present case, therefore, the two integrals of the first order in (i) must collectively vanish, while the three integrals of the second order must become collectively positive. 25. We have dV _ Vi+ y dV _ y' d'V _ 3Vi+/ ' dy ~ 2y^ ' dy' ~ ^{^ -\- y''')y df ~~ 4/^ d^v ^ y - ^r^ I dydy' 2 \/\i +/V dy"" |/(i 4-/7/ Hence equation (i) becomes A — Sy \ dx. (2) 24 CALCULUS OF VARIATIONS. Whence we have Now, it might at first appear that we could regard 8y and dy' as independent, and thus might equate to zero each of the integrals in (3). But since the curve is to pass through two fixed points, this condition, which has not yet been regarded, must be imposed upon the problem, and may be said to limit, in some sense, the independence of ^y and Sy\ This condi- tion can be imposed by means of the second integral only, since the first is incapable of any further integration. For putting for dy' its value from (A), we have 4/(1 +y>^-^ ^(i+y> ^ dx ^{^ij^yy •" Hence, since Sy^ and Sy^ are zero, when we make the integral definite, the two terms which will be without the sign of in- tegration will disappear, and we shall have and therefore (3) may be written _rJi^L3^ + -^ _-/_ U^^.^o. (4) t/o^o I 2y^ dx ^^{j^j^yyS Thus 6y' has been eliminated, and there being no further conditions to impose; (4) can only be satisfied by writing ^•+^%^ -..4=^ = 0. (5) 27^ dx ^i^iJ^yy Multiply the first term by dy, and the second by its equal, ydxj and we have PLANE BRACHISTOCHRONE BETWEEN TWO POINTS. 2$ ^=^dy-\-y'- — •"- dx = o, (6) Then by parts, and again by parts, so that we have, finally, — -^ T^~ — a constant, say -—^, if) |/(i +y^)j/ y ■ ^ Va . ^^ Now reducing the first member to a common denominator, Ave have = ^, y{i+y^) = a, y^ = ^^; (8) 1/(1 +y^)j^ |/^ J which last equation cannot be integrated by solving for y. But we readily obtain I dx Vy -7 or -— = - — -•^— , ji/ dy s/a—y which is as before the differential equation of the cycloid, in which a equals 2r ; only x and y have been interchanged, as will appear from equation (lo). Art. 17. 26. If we disregard the condition that the curve is to pass through two fixed points, we shall have, from (2), 26 CALCULUS OF VARIATLONS. Now the first of these equations can only be satisfied by equating to zero the coefficient of dydx, and then, as we may evidently neglect the supposition that y is infinite throughout the curve, we have, necessarily, s/i +/' = 0, y=± V^^ ; a result which shows that a solution by this method is impos- sible. The solution y = o of Art. i8, which will become in this case J/' = CO , is also suggested by this method ; for if in the second of equations (8) we make a infinite, then, since y cannot be always infinite, we shall find that y is infinite. This solu- tion, representing the vertical through A^ has been already shown to give a true minimum ; although the considerations of Art. 20 show that it could not be investigated so long as the horizontal is taken as the independent variable. This case then exemplifies the remarks there made relative to over- coming, by a change of the independent variable, the difficulty presented by the occurrence of infinite quantities. 27. Let us now examine the sign of the terms of the second order in dU. Since those of the first order vanish, we have, from (2), ^-'£■1 sy -^2x^1 +yy H /. , ,^^ ^ c dx, (10) From the second of equations (8) we have |/yT + yr)= ^a, PLANE BRACHISTOCHRONE BETWEEN TWO POINTS. 2y and therefore (lo) becomes 6U =S; i 'l§Sf - -4. 6y Sy + -^ Sy^ I d.. (I I) ^^° ( oy 2y Va 2a Va j But ?— can be written ^ , where r = -, or the radius 8/' 2 Va . 2/ 2 of the generating circle. Whence (i i) becomes 6U=-^r\^,S/-^-dy 6y' + I dy'^ \ dx. (12) But, from equation (A), // Sy6yd.=l'-l-r-l4.y-.cl.. (13) «/j-^-^ y 2 ^ 2 dx y ^ Put/ for •^'. Then y £y ,y Sya, ^ 1 [i^ s/, - /, Sf] - j/r^/^^^. (14) But since the extreme points of the curve are fixed, dy^ and 6y^ are each zero, and we have £y sy sydx = - '-Xyy ^ dx. (IS) But ^dx^di^ y^y' -/'^y ='^-t-yLdx; dx y y y^ and because dy = y'dx, the last equation may be written ^^ ^^ - j I y'dy y dx \y dy f ^^dx=\iy^-Qdx. (16) [y dy f ) 28 CALCULUS OF VARIATIONS. Now differentiating the third of equations (8) and dividing by 2, we have , .r ^ 2/ f and therefore (i6) becomes dl , — dx=- dx -\-+'~ \/^/ • dx. But, from the third of equations (8), whence y"-- 2r — y ~ y ' Therefore dl , --dx:= dx dx. (17) . ly^6y8ydx = \lyf^JL^dx. _ (18) Substituting this value in (12), we have, finally, and it is evident that this integral is positive, since each of its elements is positive. 28. Although we might infer from the preceding article that we have a minimum, that term being used in its technical sense, still our investigation will not be entirely trustworthy unless we regard the direction of the tangent at A as absolutely fixed. For we have seen that y = ^ — y ^ ^^^ therefore when y PLANE BRACHISTOCHRONE BETWEEN TWO POINTS. 2g y is zero, y' becomes infinite. That is, we cannot with confi- dence include in our investigation every element of the definite integral U, because at ^, F becomes infinite. We cannot, however, conclude that there is not a minimum, because we do not know what effect a variation of y in this element would have upon the general result. Indeed, we do know that if the second point be not beyond the vertex, we have a true mini- mum, and we now see also that if the tangent at A be fixed — that is, if the cycloid be compared with any other derived curve whose tangent is at right angles to the horizontal — we shall in any case have a minimum. The term derived will be used to denote any curve which can be obtained from the original or primitive curve by the method of variations, and must therefore be always indefinitely near to its primitive, and without abrupt change of direction. 29. The preceding discussion shows the advantage of taking the vertical as the independent variable. For while the result by either method is the same, as indeed it must be in every case, it is much more easily obtained by the former. This is due to the fact that in the former case x, being inca- pable of variation, enters the function F, thus leaving y only to be varied, while in the latter y and /, both being capable of variation, enter V, thus rendering the problem one of two variables. When we come to the terms of the second order, the results apparently agree also. But while that in the former case is readily obtained, and is probably entirely trustworthy so long as we do not wish to pass the vertex, in the latter case some transformation is required in order to obtain any result, and even then, owing to the occurrence of an infinite value of y at the outset, we cannot rely implicitly upon our investigation unless we regard the derived curve as having at A the same tangent as its primitive ; that is, the vertical. 30 CALCULUS OF VARLATIONS. Problem III. 30. // is required to determine the form of the plane curve which shall pass through two fixed points^ and which shall include between itself ^ its evolute, and its radii of curvature at the two fixed points a minimum area ; the extreme tangents of the required curve being also fixed. As before, let ds be an element of the required curve, r the radius of curvature, and U the area which is to become a minimum. Then U=f\ds, (I) and we must first express U in terms of x, y, y, etc. We have ds = Vi +/' dx, ^_ (dx^ + d yy^ d/ _ ds^ _ V{i+yy ,. dxdy dxdy y"dx' y" ' ^^ the sign ± having been disregarded. Substituting these values, and assuming that the curve is to be concave to the axis of X, and y" therefore negative, (i) may be written Now change y' into y' -f- Sy\ y" into y" -\- Sy\ and develop as before. Then including the terms of the second order, we have 2t/xo ( y y V^^^^f^]^^. (4) THE E VOLUTE PROBLEM. 31 Whence we must have Now it is plain, as before, that the two integrals combined in the last equation are not independent, there being here two conditions to be imposed upon the problem ; namely, that ^}\ and ^y^ shall vanish, and also that Syl and Sy^ shall vanish. To impose these conditions, we have only to extend the method already employed. Thus, putting K for -^ ^ „ -, we have / K Sy'dx := K dy — I "--—- 6y dx, /;■ K Sy-d. = /r. 6y, - a; Sy„ - ly^Sy dx. (6) Also putting L lor ^ „•; ' , and observing .that y ^ ~ dx' ~ ~dx ' we have And in a similar manner we obtain r 'jL Sy'd. = m 6y^ - m] 6y^ - T' ^ 6y d.. 32 CALCULUS OF VARIATIONS. Collecting and arranging these results, (5) becomes Now, if we suppose Sy^, dj//, ^j/^, Sj//, to severally vanish, we shall thereby impose the two given conditions upon the problem, and (8) will become As there are no further conditions to impose, this equation can only be satisfied by writing Restoring the values of K and Z, and integrating, we have y" ^ dx f' -Vc-o, (II) which, since dy' z=z y"dx, may be written ^l(LpylUy^y,,Sl±^ + cdy' = o. (12) Then integrating by parts, we have THE E VOL UTE PROBLEM. 3 3 Hence (12) gives iSL^yy^cy+c'=.o, (13) (I +yy ^ 4; But from equations (i), (2) and (3) we readily obtain r _ dx and substituting this value in (14), observing t\i2itydx = dy, we have Let t denote the angle which the tangent to the required curve at any point makes with the axis of x. Then ~ = sin /, dx and -J- = cos t. Also, let d be the constant angle whose ds natural tangent is -. Then c = h sin b, and c' = h cos b ; // being some constant at present unknown. Then substituting in {15) these values of c, dy ~, — -, it becomes ds ds r =. — (sin / sin ^ -f- cos t cos ^) = - cos (/ — B), (16) which is the intrinsic equation of the cycloid, h being equal to eight times the radius of the generating circle, and b the angle 'made with the axis of x by the chord joining the cusps. 31. Let us next examine the sign of the terms of the second order. Since those of the first order vanish, (4) becomes 34 CALCULUS OF VARIATIONS. - 4/ ^- Sy'Sf + (' +/")' y- } dx But since the axis of x is so taken as to render the cycloid concave to it, y" is always negative, and therefore the factor — ~ is always positive, since the sign of each element depends y upon that of this factor. We infer, therefore, that the cycloid is the curve required ; although, because y" becomes infinite at the two cusps, our investigation will perhaps be subject to some doubt if we are obliged to include either cusp within the range of integration. 32. If we attempt to neglect the two conditions which are to hold at the limits, and to regard 6y' and Sy" as independent, we shall have the two equations 4/(i +yo _ ^ (i+yT _ ^ both of which give y'^ = 00, which cannot be true if the re- quired curve is to be continuous. The seeming solution, THEORE TICAL CONSIDER A TIONS. 3 5 y' = o, of the first equation must be rejected, because, if it could hold, the curve becoming a straight line would cause j^' to vanish also, and thus the equation would become indefinite. 33. It is evident that the cycloid will not give the least possible value of the area in question. For by joining arcs of cycloids, or even of circles, of indefinitely small radius, the area may be made as small as we please, as will appear by the subjoined figures : We have, therefore, theoretically only a minimum in the tech- nical sense hitherto explained. In fact, the method here employed excludes all curves having either y or y' infinite within the given range of inte- gration ; and it also enables us to compare the cycloid with such curves only as can be derived from it by any arbitrary indefinitely small changes in the values of y and y. Still, under the conditions which we imposed upon the problem — viz., that the extreme points, and also the direction of the extreme tangents, should be fixed, and the subsequent condi- tion that the required curve should be concave to the axis of X — there can, we think, be no doubt that the cycloid gives not only a minimum, but also the least value of the area in ques- tion. Section II. CASE IN WHICH THE LIMITING VAL UES OF X ONL Y ARE GIVEN. 34. The reader having now become somewhat familiar with the general method of the calculus of variations, we shall next present some theoretical considerations, which are usually advanced before the discussion of problems is attempted. ^\ 36 CALCULUS OF VARIATIONS. Suppose we wish to determine the conditions which will render U a maximum or fninimumy where Then it will be found, as in the preceding examples, that U can be reduced to the form U : r^vdx, where V is some function of x, y, y' , y'\ etc. Now change jF intoj/ + Sy, / into/ + 6y\ etc., x remaining unaltered. Let V, in consequence of these changes, which are indefinitely small, become V\ and U become W, Then we shall have U'= r'V'dx. tlXo Also let U' - Uhe denoted by dl/, and F' - Fby ^V. Then if Sy, sy, etc., be indefinitely small, ^Uand ^Fwill also be in- definitely small. It is clear also that we shall have U' -U or dU= r^V'dx- r^Vdx = r\ v -v)dx= r^svdx, ( I) Now if we develop (^Fby Taylor's Theorem, it becomes -^1(a df J^ 2 B 6y Sy + CSy'-\- zDSy 6y" + etc.), (2) THEORETICAL CONSIDERATIONS, 37 in which — , — >, etc., are the partial differential coefficients of dy ay F with respect to y, y\ etc. ; and A,B, C, D, etc.^ are the second partial differential coefficients of V with respect to the quan- tities whose variations immediately follow them. Substituting this value of dFin (i), it becomes _|_ \fJ\A S/^2B dy dy' -{. C dy" -^ 2 D dy dy" + etc.) dx. (3) Now, by our previous reasoning, the first Integra] must vanish for either a maximum or a minimum, while the second integral must become negative for a maximum and positive for a minimum. 35. It has probably been observed that our treatment of the terms of the first order has been quite uniform, while our treatment of those of the second order has differed in nearly every case. The general discussion of this latter part of the problem, or, as it is called, the discrimination of maxima arid minima, is the most difficult of all the subjects connected with the calculus of variations. Although the foundations had been laid by Legendre and Lagrange, and the problem could be solved in certain cases, still no general method was known prior to the year 1837, when Jacobi published a theorem, which we shall explain hereafter, and which reduces this portion of our investigation also to a uniform rule. We shall, therefore, at present speak of dU 2iS involving terms of the first order only, except when the contrary is expressly stated. 36. Let us now consider more generally than hitherto the equation 6^^= o. 38 CALCULUS OF VARIATIONS, By (3) this becomes and this equation is true whether the values of y, y, y" , etc., at the limits are fixed or not, it being merely required that the limiting values of x only should be fixed. Now by means of the known relations given in formulae (A), (B) and (C) we can, by integration by parts, transform any term in (4) until it shall consist of terms free from the sign of integration, and an integral involving dy dx. Let N, P, Q, R, S, etc., be the coefficients of Sy, Sy\ Sy\ etc., in (4), and consider for example the term «y. =^, We have ^ dx' dx^ ^ dx dx^ rdSd'dy_ _ _ dSd'dy Pd'Sd'dy ^ "J dx 'dx' '~ dx dx" "^^ dx' dx" ' rd'Sd^ ^ _ cPSdSy _ rd'Sddy ^^ J dx"" dx"" ~ dx' dx ^ dx' dx ^ dx' dx dx' -^ ' ^ dx' rs6/^d.=(ssr-f6y"+gsy-g,sy xq -^ \ -" dx dx"" dx' + / -r-r ^y dx. THEORETICAL CONSIDERATIONS. 39 Integrating the other terms in (^^ in a similar manner, collect- ing and arranging the results, we have ^^=r-^+^^--^3-+etc.w /^ dQ , d'R d'S . \ . , /^ dR . d'S ^ \ . , (^ dR . d'S M . / -[^-d^ + d?--''''l'^^ +{^-S+^4"-^^^^-(^-S+^^^-)^^»'' + (S- etc.), c^j^/" - (S - etc.)o <^Jo"' + etc. in which -j-, -y^, etc., are the total differentials of these quan- dP (PQ dx'' dx^ titles with respect to x. Finally, for convenience, (5) may be written thus SU=L-£^'Mdydx, (6) and this equation holds, whether the values of Sy^ 6/, dy" , etc., at the limits vanish, as we have hitherto supposed, or not; the limiting values of x only being required to remain fixed. 40 . CALCULUS OF VARLATLONS. 37. We see that SU\n (6) consists of two classes of terms which are essentially different ; the first depending solely upon the values which the quantities (^J/, ^y', etc., and P, Q, R, etc., with their total differential coefficients, may have at the limits ; while the second is an integral involving the general values of these quantities. Now since (5^ t/ must vanish when ^is to be a maximum or a minimum, let us consider these two parts of (^ ^ separately in this case. Write, for convenience,. L = h, dy- K ^jo+ \ ^y:- k h:-\-j. h:-j\ ^y:+ etc. (7) Then it is plain that the several quantities Sy^^ 6y^, Sy^, Sy^\ etc., are entirely in our power ; that is, we may impose at the limits any conditions we please, so long as all the variations are indefinitely small and x^ and x^ remain immutable. It is likewise clear that the quantities h^, h^, i^, i^, etc., are not in our power. For suppose the equation dU= L -{-fj Mdydx = (8) to have been solved so as to give j/ as a function of x, say/(;ir). Then this equation would be a solution of the problem to find the value of y, or the equation of a plane curve, which would render Udi maximum or a minimum ; and as we wish to compare only this primitive with its derived curves, we must consider ^0, //j, etc., as referring to this primitive and to the given limits only. These quantities can therefore, so soon as the equation of the curve and the values of x^ and x^ are known, be found. Hence if h^, h^, i^, Zj, etc., do not severally vanish, we can make L assume any infinitesimal value we please by suitably choosing dy^^ dy^, Sy^\ dy^\ etc. But if the solution y =z f[x) cause these quantities to severally vanish, L must become zero also, and no other condition will cause L to vanish necessarily without restricting the values of Sy^^ 6y^^ dy^', etc. THEORE TICAL CONSIDER A TIONS. 4I 38. Let us now consider the second term, £y'y dx. In this integral Sy is wholly in our power, being subject only to the condition that neither it nor any of its differential co- efficients, to the 7tth inclusive, shall become appreciable within the range of integration, y"^^ being the highest differential co- efficient in V. In other words, Sy may be any arbitrary func- tion of X which fulfils these conditions, or it need not even be the same function throughout the entire range of integration. On the other hand, M is not in our power, but will, as in the case of h^, h^, etc., depend upon the equation j ^=zf{x). Hence if M be not necessarily zero throughout the given limits of integration, the integral will be wholly in our power, and we may, by suitably varying y, make it assume any infinitesimal value we please. But if the solution y = f{x) reduce M to zero throughout U, then the integral itself, being definite, must become zero ; and it will not necessarily vanish under any other condition, so long as 6y is wholly unrestricted. 39. Resuming equation (8), Ave have L=-S.ySydx. (9) Now if the solution j =/(;i;) be such as to cause the quantities /^o, h^, i^, /„ etc., and also M to severally vanish, then each member of (9) will likewise vanish, and no difficulty will occur. But if the proposed solution be not able to fulfil all these con- ditions, (9) becomes an impossible equation. For inasmuch as L and J^ M dydx are no longer necessarily zero, it would in effect imply, as Prof. Jellett has remarked, " that the inte- gral of an arbitrary function may be expressed (without deter- 42 CALCULUS OF VARIATIONS. mining or even restricting its general form) in terms of the limiting values of itself and a certain number of its differen- tial coefficients. This is manifestly untrue." We conclude, then, that it is necessary to the existence of a maximum or minimum not only that L and M shall vanish, but that each of the quantities h^, h^, z^, z^, etc., and M, shall become zero. 40. Although the truth of the preceding principles would appear to be sufficiently evident, yet Strauch, one of the most elaborate writers on the calculus of variations, asserts that it cannot be proved that L and / M dy dx must severally vanish ; and as this is a point of the highest importance, and of some difficulty, we have given it more attention than it has generally received hitherto. Strauch is, however, compelled to admit that we do obtain correct results by this method ; and there can, as Prof. Todhunter states, be no doubt that the principle is sound. 4-1. Before proceeding further we will apply the foregoing theory to the solution of some examples. Problem IV. Let V be any function of y" and constants only, and let it be required to detennijie the relations which vizist szibsist between x and y in order to maximize or minimize the expression u=syd^^ x^ and x^ only being fixed. We have SU=^ r^^^-^r h"dx = r^Q S/dx = o. THEORY ILLUSTRATED. 43 Then transforming dU, as just explained, and denoting by accents total differentials, we have +ir' Q'Sydx^Q, (i) Whence, since M must vanish, we have Q'^o, Q = c, Q = cx + c'^ (2) If we had supposed the values y and y' at the limits to be given as in former examples, the solution could be carried no further without determining the form of V. But since dy^, ^y^ ^Jo^ ^yl ^^^ ^^^ necessarily zero, we must, from the pre- ceding discussion, have the coefficients of these quantities severally zero. Hence 2/ = o, QJ = o, Q, = o, Q^ = o. From the third and fourth of these equations, combined with (2), we have cx^-{- c' = o, ex, -\- c' = o, c{x^ — x^ = o. Whence c = o, and then c/ = o. Therefore the last of equations (2) gives Q = o. If this equation is to hold throughout U, y" must be con- stant, although it may have several constant values. Let a be one of the roots of the equation Q — o. Then, as y" = ^, by integration we obtain j/=^ + ^^ + -5', (3) the equation of a parabola ; or of a straight line if a should happen to become zero. The constants b and b' cannot be determined so long as the values of j^ and y^ are not fixed. For it is easy to see that the equations Q' = o and QJ = o furnish no new equations 44 CALCULUS OF VARIATIONS. of condition, because they follow from Q — o^ and any values of b and b' which satisfy the latter will also satisfy the former two. Owing to its simplicity, we may also examine the term of the second order^ which is 2«^^o dy Since y" is a constant, ~^, which is some function of y\ dy must be also a constant, say A ; then, since the terms of the first order vanish, we may write 2 ^^ which shows that we have a maximum or minimum accord- ing as A is negative or positive. Problem V. 42. // is required to maxiinize or minimize the expression the limiting values of x only being given. We have SU= r\^y"^y" - 2^J) dx = - 2yl" Sy^ + 2y:" dy^ + 2J'." &J'/ - 2y: 5y: +fj\2/-' - 2) 6ydx = o. (i) THEORY ILLUSTRATED. 45 Whence equating M to zero, and integrating, we have y'=i, (2) y^x + a. (3) y^'^-^ax + b, (4) y^^ + '^ + ^^+c, (5) X'* . ax^ \ i>x^ , , , ^ = 24+6 +2 +^" + '^- (6) Now if dy^ and Sy\ be unrestricted, we must have, from (i), y^" — o,y^" — o, which give, in (3), x\-\-a= o, and x^-\-a-=o, which are impossible equations, since x^ and x^ are not to be equal. Whence we conclude that the solution will not be possible unless we restrict Sy^ and Sy^, so that y^" and y^" need not severally vanish. We will now suppose y^ and j, to be given, but j/ and j/ to be unrestricted. Then, from (i), we must have y" = o, y^' = o, which would give, in (4), ^^ax, + d = o, (7) x: 2 -J^ax, + d = o, (8) From these equations we readily obtain ^ = -^-(^o+^^)y (9) 2 i = ^. (10) 46 CALCULUS OF VARIATIONS. Now suppose, for simplicity, that we take x^ equal to any constant e, and x^ to — e. Then (9) and (10) give ^ = o, b = , and (6) would become x" e'x'' y = \-cx-\-d. (11) 24 4 ' ' ^ ^ But it will be remembered that this equation is only admissible on the supposition that we are able to make h^^y^ — hfy^ vanish ; and as h^ and h^ cannot severally vanish, this is accomplished by fixing the values of y^ and y^, and the assignment of these values will afford us the conditions for determining the remaining constants. Equation (11) now gives -^f+ ce + d, (12) '■---^- ce^d. (13) Whence we obtain ,_yr-yo '- 2e ' (14) "=-+£ 2 (15) Suppose, for still greater simplicity, we take the fixed points on the axis of x. Then (14) and (15) give c = o, d = ^—, and we shall have, finally, 24 _x' _e^\5/ -^""24 4 24* But suppose, as usual, the limiting values of y and y were both given, and let us consider the particular case m which THEORY ILLUSTRATED. A,7 we have x^ = e, x,=^ — e, y, = o, y^ = o, y/ = i, yj = — i. Then, from (5), we have Then eliminating c between (17) and (18), we have and from the same equations we obtain ^ = -J-' (20) Moreover, substituting in turn e and — e for x in (6), we have ■^■ = ^ + T + T + ^' + '^' (^') Eliminating dy we have — + 2^^ = o. (23) Substituting for a its value from (20), we find ^ = o, whence also a = 0; and again substituting these values with that of b, (21) gives 24 2 48 CALCULUS OF VARIATIONS. Now substituting these values in (6), we have, finally, y— x" -\ . 24 \2e 24 2 The term of the second order is merely which is of coUrse positive, thus giving a minimum. That is, any solution which reduces the terms of the first order to zero will render U a minimum.^ 43. Now resume for a moment the consideration of Prob. I. There we have which give y^ = o, and 7/ = o. But since we know from the general solution that y' — a, these two conditions are in reality only one, a =^ o. Hence if no restrictions be imposed except that x^ and x^ shall be fixed, the line must be parallel to the axis of X, But the constant b cannot be determined in this case. In- deed it is evident that the straight line parallel to x is shorter than any other curve, or straight line even, which can be drawn having x^ and x^ as the abscissae of its extremities, and that hence our first result is confirmed. Moreover, since the length of this line will be the same, whatever be its distance from the axis of x, the value of b can have no effect upon its length, and therefore ought to remain undetermined. If, however, the co-ordinates of one of its extremities be given, the line becomes a parallel to x through that fixed point, and b is determined. * The last two examples are from the Adams Essay, by Prof. Todhunter (p. 15), but have been considerably elaborated. THEORY ILLUSTRATED. 49 4-4. Next consider Prob. II., Case i. There we find But from equation (8), Art. 17, if we make h^ or h^ zero^ we see that the equation y — Q must hold throughout the curve, and this gives y' = o, which, as it denotes the vertical, is the true solution. For if a par- ticle be merely required to descend from one horizontal plane to another, it will do so along the vertical sooner than along any other line. The equation of this vertical is y = d, in which the value of d can have no effect upon the time of descent, and therefore remains undetermined, as it should. Next consider the second case of the same problem'. There we have The first of these equations gives j// = o ; and since V{i -\-y''')y — V2r, r being the radius of the generating circle, we have .( |/(i+y>f „ V2r and this, if equated to zero, will give // = o, which is evi- dently impossible. Hence h^ cannot be zero ; and to make the term h^Sy^ vanish, we must assign the value oi y^. Now it will be remembered that the general solution was a cycloid, having a cusp at the starting-point of the particle, and that b was merely the value oi y^, which is now deter- mined. Moreover, since we have just found that the tangent to this cycloid at the point which is not fixed must be par- 50 CALCULUS OF VARLATLONS. allel to the axis of x, it follows that its vertex must be at this point. Hence the generating circle must be such that it would roll through a semicircle while its centre was de- scribing the distance x^— x^, and therefore we have ^1 — ^0 r = ~ -. 45. Let us in the last place consider Prob. III. If we could have fully integrated equation (lo), Art. 30, the inte- gral would have involved four constants, and for determining these constants we would have 7/, y^, j/^, y^ equal to four assigned quantities. It would, however, be too tedious to discuss this case in detail, and we will next suppose the values of 7o and y^ to be fixed, while those of j// and 7/ are variable. Then equating to zero the coefficients of ^yl and Sy^^ we shall have ^l.= and since \/ i-\-y''^ cannot be zero, y" and y" must each be infinite, thus giving the cycloid cusps at the two fixed points. Let b denote the angle which the line joining these cusps makes with the axis of x. Then b is identical with b of equa- tion (16), Art 30, and is at once determined, its tangent being y^ - y. ■^1-^0 Then, also, 8 27t Let us now suppose the values of y^ and y^ to be unre- stricted. Then we must equate the coefficients of ^y^ and djo severally to zero, which will give the equation THEORY ILLUSTRATED. 51 and. a similar equation for the lower limit. But from equa- tion (it), Art. 30, the first member of the last equation equals — c, making c in this case zero. Therefore equation (13) of the same article becomes (i+yy _ d y" 2* Now d cannot vanish. For if it can, we must either have \/\ 4-y = o, which would render y imaginary, ox y" must be infinite throughout the curve, which is also inadmissible. But if dy^ and Sy^ do not vanish, we must, as we have just seen, have y" and y^' infinite. It follows, therefore, that j// and 7/ must become infinite, as d would otherwise vanish. We conclude, then, that the cycloid must in this case be so placed as to have the line joining its cusps parallel to the axis of X. Then we shall evidently have ^ ^ ^, - ;r. 27r while the constant angle b of equation (16), Art. 30, will be- come zero, and it is easy to show also that <^ = 8r = h. 46. It is evident that none of the results of the preceding articles could be confirmed as maxima or minima without an examination of the sign of the terms of the second order, because even if those terms were shown to be certainly posi- tive or negative, in any particular problem, by making any of the variations Sy^, Sy^^ 6y^', dy^', etc., zero, it would not fol- low that we could be certain of the same sign when those restrictions were removed or modified. But it will be remembered that in the problems thus far discussed we have, with the exception of Case 2, Prob. II., been able to determine the sign of the terms of the second order without imposing any restriction upon the variations of y and y' at the limits. The only result, then, which we 52 CALCULUS OF VARIATIONS. have to confirm is this : that when the starting-point of the particle is given, its terminal point being restricted to have a given abscissa x^, the curve of quickest passage from x^ to x^ will be a cycloid with a cusp at the first point, and its vertex at the second. An examination of equations (12) and (13), Art. 27, will show that if we had not supposed Sy^ and dy^ to be zero, equation (19) of the same article would have become 27 in which the integral is positive as before. But by hypothesis dj/„ = o, and asy vanishes at the vertex, while j^ becomes a or 2r, we have (— ) = o. Hence both terms without the sign of integration vanish, and we have a minimum as before. 4-7. We may now proceed without difficulty to that gen- eral discussion of the terms of the first order which is usually, but unadvisedly we think, presented prior to the discussion of particular problems. Assume the equation U — I Vdx, where V is any func- tion of X, y, y' . . . . y^\ and let it be required to determine what function y must be of x in order to render U a maxi- mum or minimum. Then finding dU, and transforming it by integration as far as possible, and then equating to zero sever- ally the coefficients of dy^, dy^, etc., together with M, which is the coefficient of Sy dx under the integral sign, we obtain the equations h^ = o, h^ — o, i, = o, z^ = o, etc., and also M = o, where, as will be remembered, ,, dP , d'Q THEORETICAL CONSIDERATIONS RESUMED. 53 all the differentials being total, and iV, P, Q, etc., being the partial differential coefficients of V with respect to y, y\ y", etc. Now the equation M = o will, in general, be a differential equation of the order 2;/, because its last term will be d^ dV which will usually involve dx' = (yw))(«) = y {2ny Hence the complete integral of this equation must usually contain 27t arbitrary constants, and may be supposed to be put under the form y = f{x, C„ C„ c,n) = /. (i) Now since every solution of our problem must satisfy the equation M=o, it must also be comprised in (i), which establishes a general relation between x and y, or, in other words, gives us some plane curve ; which relation or curve is, however, capable of great modification, by adjusting suitably the values of these 2n arbitrary constants. 48. If now we examine the equations //, = 0, ^0 = o> ^tc, which we may call the equations at the limits, we shall find that their number is also 271. Moreover, these equations, not holding throughout the curve, do not establish any general relation between x and y, as did the equation M=o, but merely fix the conditions which the required curve must fulfil at the limits. This is as it should be. For if the equations h = o, i — o, etc., could be supposed to hold throughout the curve, they would each establish a relation between x and y, and unless these relations should happen to agree with each 54 CALCULUS OF VARLATLONS. Other, and also with that derived from the equation M = o, which would seldom if ever occur, the solution would become nugatory. Now suppose the complete integral of the equation M =o were obtained, and expressed as in (i). Then if the form of/ were known, we could form the expressions h^, h^, t^, etc., and these expressions would all be known functions of either x^ or x^, together with some of the 2n arbitrary constants, no vari- able entering these functions, because x^ and x^, being assigned quantities, may be regarded as constants also. We see then that in the equations h^ = o, k^ = o, etc., we have 2n equations between x^ and x^ which are assigned, and 2?t arbitrary constants, and should therefore be able to deter- mine these 2n constants in terms of the known constants x^ and x^. Now suppose the limiting values of y^ and y^ were given. Then, since the variations of these quantities would become zero, /^, and /i^ would no longer necessarily vanish. But in this case it is evident that the two equations thus lost would be replaced by the equations jk^ — f{x^, c^, c^ . . . . c^n) —fv ^nd y. = A^o, c,,c^ c^y^ = /„ ; and as jk^ and y, are now sup- posed to have assigned values, the number of the equations for the determination of the 2n constants remains, as before, 2;/. In like manner, if ^// and Sj/J should become zero, the con- ditions i, = o and t, = o would disappear. But to supply their place we would have the equation and a similar equation for the lower limit, j// and j/ being now assigned constants also ; so that we still have, as before, 2n ancillary equations. Suppose, lastly, that any of the variations Sy^ ^y^, Sy^', etc., were connected by given equations, and suppose there were m such equations. Then if we should express as many of the EXCEPTIONS TO THEORY. 55 variations as possible in terms of the remaining variations, and then equate to zero the coefficients of the several varia- tions in the reduced system, it is plain that our ancillary equations would be only 2n — m in number. But since we have the m equations between certain variations, we are evi- dently able to form new systems of independent variations in ' such a manner as to obtain 7n more equations between x^, x^, and the 2n constants. Thus we see that, theoretically at least, the terms at the limits furnish us with 2n equations for the determination of the 2n arbitrary constants, which would in general occur in the complete integral of the equation M—o, and that what- ever condition reduces the number of the original equations, by annulling or combining two or more of them, will at the same time furnish in their place as many new equations for the determination of these constants as have been removed. 49. The preceding considerations, which are theoretical, require some modification, first as regards the terms at the limits, and second as regards the equation M = o. With regard to the terms at the limits, it has probably been noticed that it has not been in general possible to satisfy all the equa- tions //, = o, //„ = o, etc., as some of these equations become conflicting. But even in these cases we can, as we have seen, generally obtain 2n harmonious equations by restricting one or more of the variations ; as, for example, by supposing dy^, dj/^, or (^K/, etc., to vanish. In fact, the occurrence of these conflicting equations de- notes merely that the problem in its present form is not capable of solution, and as it might be foreseen that such questions would present themselves, the occurrence of these conflicting equations would naturally be expected. 50. The following exceptions may be regarded as due to the nkture of the equation M = o, although they properly arise from the nature of the function V. 5^ CALCULUS OF VARLATIONS. Exception i. Suppose N to vanish in the equation M = o, which would of course happen if y did not exphcitly enter V. Then we would have whence dx^ dx' ^^'^•-O' ax dx But the first member of the last equation equals h ; and as h must vanish at either limit unless the values of )\ and y^ be assigned, we have c =.o\ and since the equations h^ = o and //o = o are each satisfied by this value of c, they furnish no new condition for the determination of any other constant which may enter the complete integral of the equation M = o. Thus the conditions furnished by the terms at the limits are in this case reduced to 2;^ — i, two of them having become identical. If, however, the value of either y^ or y^ be assigned, this will furnish a new equation of condition which will compensate for that which was lost. This case is fully exemplified by the discussion of Prob. I. in Art. 43 and Prob. II., Case i, in Art. 44. Similarly, suppose V to contain neither y nor y'. Then we would have ,^ d'Q d'R , ^ ^ ' ^ dO d'R dx dx^ etc. = a, (2) G-g + etc. = «^ + ^. (3) Now if the limiting values of y are variable, we have h^ = o and //„ = o ; and it is easy to see that in this case, as P is want- EXCEPTIONS TO THEORY. $7 ing, the first member of (3) is z, and that of (2) is — A, and therefore we have ax^ -J- <$> = o, and ax^ -f- ^ = o, whence we find a — o and d = o, and (3) becomes Now it must be remembered that this equation has been de- duced solely from the conditions z, — o and i^ = o. But dif- ferentiating (4), we have dQ d'R , ^ J -^ -^ + etc. = 0, QT —h= o. dx dx Whence it appears that since the equations h^ = o, //„ = o, can, without involving any other relations, be deduced from the equations z^ = o, z^ = o, they furnish no new data for the de- termination of the constants which will be found in the com- plete integral of the equation M = o. Hence in this case our ancillary equations will furnish but 271 — 2 distinct conditions, thus leaving generally two constants undetermined, unless one or more additional equations be supplied by assigning the values of one or more of the quantities j/j, jo, j/, y^\ etc. This case is fully exemplified in the discussion of Prob. IV. Generally, if the first in of the quantities y, y', y", etc., be wanting in F", while at the same time the variations of these quantities at the limits remain unrestricted, in arbitrary con- stants in the general solution must also remain undetermined. 51. Exception 2. Suppose V to contain only the first power of y^^ the highest differential coefficient which is in- volved. Then in this case the equation M=o cannot be of an order higher than 2n — i. For the last term in M must be ± -z-^ ~rj^y ^^^ ^^ ^"^^ *^^ ^^^^ power of y^) occurs in V, the 58 CALCULUS OF VARIATIONS. partial differential coefficient of V with respect to y*^> will not contain that quantity at all. Whence it is evident that M cannot be of an order 2n ; and indeed Prof. Jellett has shown that it cannot in this case rise above the order 2n — 2 (see his page 46), but it does not seem necessary to reproduce his proof here. Now in this case the equations at the limits will be, as before, 2n in number, while the constants in the complete in- tegral of the equation M —o will not exceed in number 2n— \, and in fact will not exceed 2n — 2. This seeming exception is, however, explained by the fact that in all such cases the integral U^ or J Vdx, is capable of being reduced by integra- tion to the form ^==/i — /o+ / V'dx^ where f and /, are t/ Xq quantities free from the sign of integration, while V^ does not contain any differential coefficient of y higher than y^ - ^) ; and we will next show that this reduction can be effected. 62. Let y^-) be the highest differential coefficient in V. Then, since its first power only occurs in F, we may write j7_^yn)_|_^^ (l) where w is that part of F which is a factor of y*^\ and z the other terms of V, both being of course of a lower order than y^^\ Then the equation U — J^' Vdx becomes nx-^ nxi U—J wy^'^^dx-^J zdx. (2) But we are evidently able to form the following equation : fwy^^)dx = W+fzdx, (3) EXCEPTIONS TO THEORY, 59 where Wand Z are functions at present unknown. For this equation can, if in no other manner, always be formed thus : / wy^'^^dx — wy^'^^x -\-J — — - zvy^'^'^.xdx. (4) But W and Z can be so taken that the second member of (3) will contain no higher differential coefficient than y^--^), be- cause (3) can, in the following manner, be satisfied upon this assumption. First differentiate (3), and we shall have ,, ^ , dW , dW , , dW „ , ^ , dW ,, , , ^yin^ = Z-Y^^+^^-y+-^^ry + ^tC. + ---/n,. (5) which must be the complete differential of (3) if our assump- tion be true, but not otherwise. But (5), and consequently (3), will be satisfied if we put „ dW ^ dW , ^ ^ , dW , ,. - ^ = ^- + -dy-y + ^^" + ^y^)-^'^" • W Therefore ^is found by integrating w with respect to y^^-i) only. Hence, finally, we have* = W,-W,+ r'v'dx. (8) This case, then, is in reality no exception at all, because the * This theorem is due to the great Euler (see Meth. Inven., pp. 62, 75), and has been nearly reproduced by Prof. Jellett on his page 46. 6o CALCULUS OF VARIATLONS. difficulty arises merely from the fact that the original integral had not been reduced to its lowest terms. For although we have not yet considered the class of problems to which this reduced form of U belongs, it is easy to see that the equation M = o, resulting from V^ only, will not now be of an order ex- ceeding 2n — 2, which is the result obtained by Prof. Jellett. 53. Exception 3. Let V be of the form y/-\-F, where / contains only quantities incapable of variation, e.g. x and con- stants, and F may contain any quantities except j/. Then JV becomes simply /, and the equation M = will give the equations dx dx^ etc. =r/. Now the first member of (i) equals ]i\ and if we suppose J/, and /o to be unrestricted, we must have h^ —o, i^ = o ; and using these restrictions, (i) will give {/WJ +^ = 0, (2) and '■/Or)} +. = 0. (3) But as the first members of (2) and (3) contain only one indeterminate constant, c, it will in general be impossible to satisfy both equations, and the problem in this form does not usually admit of a solution. But if we make / zero, so that V is any function not containing /, the problem becomes a case of Exception i, and may or may not, according to its nature, be capable of a general solution, one constant at least GENERAL FORMULA, 6 1 remaining undetermined. This exception is exemplified by Prob. v., in which /= - 2, F = y"\ 64. It is now evident that if we require that U shall be a maximum or minimum, the calculus of variations will ter- minate its aid in the discussion by leaving us with a series of differential equations, that of the highest order holding true for all values of x from x^ to x^, the others merely holding at the limits of integration. From the former of these equations, as it is general, the general solution must be obtained, and then the remaining or ancillary equations, not being general, mpst be satisfied, if they can be satisfied at all, by the assign- ment of suitable values to the constants which will occur in the general solution ; or we may say that these ancillary equations determine the values of the constants. The determination of these constants is not in general dif- ficult when the complete integral of the equation is known ; but this integral is often obtained with difficulty, and is some- times altogether unobtainable. In fact, this difficulty is anal- ogous to that which is frequently experienced in solving the final equation or equations of condition given by the differ- ential calculus in the discussion of an ordinary problem of maxima or minima, except that in the former case the final equations are differential and must be solved by the calculus, while in the latter they are algebraic and must be solved by the theory of equations. 55. We shall next proceed to establish some principles re- garding the integrability of the equation M = o, and to deduce some formulae which will be found useful in our subsequent discussions. Suppose, in the first place, that the first m of the quanti- ties N, P, Q, R, etc., were wanting in the equation M = o, which would of course happen if the first m of the quantities J> y'l y" 1 y'" y ^tc, were wanting in F; then the equation J/= o can be integrated at least m times. 62 CALCULUS OF. VARIATIONS. For let m be 4, for example. Then we would have which, being integrated four times, becomes S V- etc. = -—A \- ex -\- d, dx 62 and similarly if m were any other number. 56, Suppose, in the second place, that the independent variable x does not occur explicitly in V\ then the equation M= o can be integrated at least once. For since V does not contain x, we have dV= Ndy + Pdy' + Qdy" + Rdy'" + etc. = {Ny' -\-Py" -\-Qy"'^Rf'^^\.Q>,dx, (i) Now substituting in the last member of (i) the value of N derived from the equation M —o, viz., jyj_dP d'Q ^^^ dx dx^ we shall have But every parenthesis in (2) can be integrated by parts. Taking, for example, the third, and recollecting that y ^ dx = dy"\ y"'dx = dy\ y"dx = dy\ GENERAL FORMULA. ^3 we have Jr^^ dx = Ry'" -J^^y["dx, rdR ,„, dR „ . rd'R „, rd'R „, d'R , rd'R ,, Hence /{^/' + /^l^.=/^y"-fy'+^y. (3) Integrating the remaining terms in a similar manner, we would have which equation is certainly of an order lower than that of the differential equation M — o. The following particular cases of this formula are given for convenience of reference : First. If F be a function of y' only, we shall have, from (A), v=c + py. (B) But since in this case F is a function of y, P must also be a function of y' ; so that (B) may be written Ay')+/p{y)--=c =/'{/), where f is an arbitrary function. The last equation can therefore only be satisfied by making y a constant, say y' = r„ which gives y=CiX-\-c^. 64 CALCULUS OF VARIATIONS. Hence if we require the nature of the curve which will maximize or minimize the expression U =^ ^Vdx, where V \s> any function of y' only, the straight line is the solution, if there be a solution ; that question being decided by an appeal to the terms of the second order. Second. If F be a function of y and y' only, (A) will still give v=c+py. (C) Third. If F be a function of y and y'^ only, then (A) will give F-.+ e/-gy. (D) 57. Suppose, in the third place, that the independent vari- able Xy and also the first in of the quantities y, y', y% etc., are wanting in F; then the equation M ^^ o can be integrated at least m -\- i times. Let in, for example, be 4 as formerly. Then the equation M =^ o, after having been integrated four times, according to the first case, and using /, q, r, s, etc., for P, Q, R, S, etc., to prevent confusion, becomes s — \- — - — etc. = ax^ -\- bx" •\- ex -\- d, (i) dx dx Also, we have the equation dV ^sdf^ -^ tdf""^ + iidy"^^^ + etc. = (^y^) + /y^) + uy^"^ + ^y«^ + etc.) dx. (2) Substituting in (2) the value of s derived from (i), we have dv = (/y^> + ^ f\ dx + (^^y *) - ^/ j'(^^) dx -j- / ^y ') + ^^ y5)j ^^ _j_ etc. + {ax' + bx' J^cx-\-d) /'^ dx. (3) GENERAL FORMULA. 65 Integrating by parts, as in the second case, we have +f{ax' -^ bx' -^ ex -\- d) f dx. (4) Moreover, the integral sign can easily be removed from the remaining terms in (4). For, by parts, we have / ax^y^^^dx — ax""/^ —J ^axY^'dx, -fiax^dx = - saxy +f6axy''dx, I 6axy"'dx = 6axy" — I 6ay"dXy r — J 6ay"dx = — 6ay\ Hence / axy^^^dx = ax^f^ — ^ax^y'" + 6axy" — 6ay' ; and in like manner we may integrate all the other terms. Thus, for example, in Prob. IV. we find, after two integra- tions of M, Q or _,=.^ + ., which, being again integrated, gives V= cxy" - cy' + c'y" + d. 65 CALCULUS OF VARIATIONS. Or, let V be a function of y' and y" only. Then, after one integration of M^ we have P f^ = <^. ax We also have and substituting the value of P from the preceding equation, we have dV = [af + Qy'" + ^yy^y which, being integrated, gives Problem VI. 68. // is required to determine the form of the solid of revo- lution which will experience a niinimiun resistance in passing through a homogeneous fluid in the direction of x, the axis of revolution of the solid. Although it is evident that the problem does not admit of a solution until some further restrictions are imposed, we shall at present merely assume that the distance x^ — x^ is given. Let ds be an element of the generating curve, pds the nor- mal pressure which it experiences in passing through the fluid, and V its velocity in the direction of that normal, or the velo- city with which the particles of the fluid are displaced by it in that direction. Then, adopting the usual theory regarding the pressure and resistance of fluids, we have pds = czf'ds, (i) SOLID OF MINIMUM RESISTANCE. 6-/ where <: is a constant depending upon the density of the fluid. Let v' be the velocity of the body in the direction of the axis x. Then and (i) becomes j,ds = cv'^^,ds. (3) Let dz be the surface of the elemental zone, described by ds. Then, since dz = 27ryds, we shall have/, Sy, - />. Sy, +fjj- ~ Sy dx I /*^1 I + -Xp<^/V^. (4) 67. Now it will appear, by reasoning in all respects similar to that which has been hitherto employed, that since dx^ and dx^, like Sy^ and ^y^, are capable of either sign, if U is to be a maximum or minimum, the terms of the first order in \pU^^ must vanish, and those of the second must become positive SHORTEST CURVE BETWEEN- TWO CURVES. 55 for a minimum and negative for a maximum. Disregarding, therefore, at present the terms of the second order, we have lSU^ = V:dx,-V,dx,+ P,Sy^-P,dy^+J^^ - ^dy dx. (5) Now it must be evident that the curve sought can be no other than a straight line. For suppose the points a and b to be joined by any curve other than a straight hne. Then even if this curve were shorter than any other Kne which could be drawn between the given curves, when one or both the ex- treme points a and b were changed, yet we know from our previous investigations that, without changing these points, it could be still further shortened by making it a right line. Whence we see that our present problem must concern merely the position which this line must have in order to render its length a minimum. Moreover, the term under the integral sign in (5) is just what it would have been had we merely re- quired the curve of minimum length between two fixed points. dP Therefore, since the right line is the general solution, — - will dx vanish, and consequently the integral must vanish, thus leav- ing us with the terms at the limits, which must also be equated to zero. This mode of demonstration will probably be most appa- rent, but the following is the true analytical method. By rea- soning similar to that employed in Art. 39 and the preceding articles, we can show that the term under the integral sign must vanish, as must also those free from the sign of integra- tion, taken collectively. Equating the integral to zero, we obtain, as before, the right Hne as the general solution, and have then to consider the remaining terms, which may be rep- resented by the equation L—o. 68. We have, then, from (5), L == V,dx, - V^dx, -^P,6y, - P,dy, = o. (6) 86 CALCULUS OF VARIATIONS. Now if the quantities dx^, dx^, 6y^^ dy^ were entirely indepen- dent, we would evidently be obliged to equate the coefficient of each one severally to zero. Then we would have four equations at the limits to be satisfied, whereas the general solution contains but two arbitrary constants, and this would usually be impossible in any problem. But in the present case we know, without further investigation, that two of these equations, V^^o and V^ = o, cannot be satisfied by any real value of y. This is as it should be. For if the quantities ^;i:,, dx^, ^jj/j, (5>o were independent, the extremities of the required curve would be entirely unrestricted, and we could have no maximum or minimum, because we could always increase or diminish its length at pleasure. But as in the present case the extremities of the required curve are confined to two given curves, we can obtain a definite result, and we now proceed to show the method of imposing this condition upon the question. 69. Let y —f{^) and y = F{x) be the equations of the two given curves, and let y be any ordinate of the required curve, and Y the ordinate of the derived curve corresponding to the same value of x. Then Y^=y^-\-Sy^, 2ind Y^=iy^-\- dy^] and let us consider, for example, the upper limit. It is evident that when we derive cd from ab, the abscissa of d will become x^ -\-dx^, where dx^ may be positive or negative, and the value of its ordinate is evidently obtained by passing along the de- rived curve from the point whose co-ordinates are x^ and F, to the point whose abscissa is x^ -\- dx^ ; that is, to the point d. Denoting then the ordinate of d by n, we have n=Y,+ Y,'dx,-^- Y," dx^ + etc. (7) Hence, substituting in (7) the value Y^=y^-\-Sy^^ and omit- ting all terms of an order higher than the second, we have SHORTEST CURVE BETWEEN TWO CURVES. 8/ n = [y-\-dy -\-y'dx + dy'dx + - y" dx^ ■ (8) But since n is an ordinate of the given curve whose equation is Y — f{x), we must have n^^ f{x^-\- dx^. Developing this expression by Taylor's Theorem, we have to the second order «=/,+/.'^-^. + j/.V-^-A (9) where /'=£' /"=S' ^-^(^-^ Combining (8) and (9), observing that_j', =/„ we have */. = (/' -/). dx, + i(/" -/). dx^ - 6y:dx, (ID) Similarly, we have at the lower limit Sy, = {F'-y'\dx,-\-^-{F"-y\dx;-6y:dx,. (10) 70. If now we substitute in (6) the values of Sy^ and dy^ just found, and set aside all terms of the second order, which must be added to those of the second order in (4), we shall, after restoring the values of V and P, have y'f'-y' ^ = ^-^TT7+^'+^ "1^^. Having thus eliminated Sy^ and Sy^, it is evident that the remaining quantities dx^ and dx^ are absolutely independent, and that we must therefore equate their coefficients severally 88 CALCULUS OF VARIATIONS. to zero. Performing this operation, and reducing, we have the equations I ■\-ylf! = o, and I + y.'F: ^ o ; equations which show that the required right Hne ab must be normal to each of the curves ff and gg. 71. Although for the sake of simplicity we have used equation (6), it is evident that the true mode of reasoning would be the following: First eliminate Sy^ and Sy^ in (4) by the use of equation (10), by which elimination we shall add some terms to those of the second order. Then, by the usual reasoning, those of the first order must vanish. But these terms will then consist of L as given in (11), together with an integral; and, by the reasoning already employed, these two parts must separately vanish. Now by making the integral vanish, we obtain the right line as the general solution ; while by making L vanish, we obtain at once equation (11), from which we derive the same conclusion as before. 72. If we use equation (6), recollecting that it is true to the first order only, we may evidently obtain the complete terms of the second order by adding to those already in (4) those which result from the elimination of Sy, and Sy^ in (6) by the use of equation (10). But these terms will by either method become, since those of the first order vanish, ^Jx, 2(1 +y^)^ -^ ^ ' Now the integral in (12) is known from Prob. I. to be posi- tive, so that we shall be sure of a minimum if the remaining SHORTEST CURVE BETWEEN TWO CURVES. 89 terms be positive, but not otherwise. But since the solution y' is a right line, — -^ is a constant, say c, and these terms 1/1 +/' become [f:dx: But c is the sine of the inclination of ab to the axis of x, and we may therefore so assume this axis as to render c positive, and then we shall be sure of a minimum if f" be positive while F^' is negative. 73. But it is unnecessary to pursue this investigation fur- ther. For it must now appear that the problem under con- sideration is one rather of the differential calculus than of the calculus of variations. For since we know from Prob. I. that the right line is the plain curve of minimum length between two points, whether they be situated upon given curves or not, we might have been certain beforehand that the solution could be no other curve than the right line, and that our problem could concern nothing but its position. Moreover, its posi- tion being determined, we need only compare the line with other right lines drawn to points on ff and gg consecutive to a and b. For if we vary ab so as to obtain a derived curve, cd, which is not exactly a right line, then, even if we show that ab is shorter than cd, we could shorten cd by making it a right line, its extremities remaining unchanged, and could not without a new comparison be certain that the new line cd might not be shorter than ab. The problem might then have been enunciated thus : To find the position of the right time of miiiimtun IcngtJi which can be drawn between two given plane curves. 74. Although problems of this sort might be altogether omitted here, there appears — at least so far as the terms of the first order are concerned — to be some advantage in solving 90 CALCULUS OF VARIATIONS. them by the calculus of variations instead of by the ordinary methods of maxima and minima. At all events, they are gen- erally discussed by writers on this subject, and it is deemed necessary to render the reader familiar with the methods which they employ. We shall therefore subjoin a few more problems of the same kind, considering the terms of the first order only, since a discussion of those of the second would in general be unsatisfactory. Problem X. 76. It is required to determine both the nature and position of the curve which will minimize the time of descent of a particle from one given curve to another, the particle starting from a fixed horizontal line, and being acted upon by gravity solely, all the curves lying in the same plane. Assume the fixed horizontal line as the axis of x, and let x^ and Jo be the co-ordinates of the point in which the required curve cuts the upper of the given curves, while x^ and y^ are the co-ordinates of the point in which it cuts the lower. Then, reasoning as we did in Prob. II., we see that we have to min- imize the expression Now it is clear that here, as in the preceding problem, the limits of integration will be also subject to variation. For suppose that after the required curve and its points of inter- section with the given curves have been found, we assume points on the given curves consecutive to those just found, and then connect these new points by another curve. Then the abscissse of these new points will be x^ + dx^ and x^ -\- dx^, dx, and dx, having either sign. It also appears, as before, that the total change which U will undergo, both from a BRACHISTOCHRONE BETWEEN TWO CURVES. QI change in the form of the curve and an alteration in the posi- tion of its extremities, can be found by first changing the Hmits of the integral in such a manner that the new limits may be the abscissae of the new points, while the form of the curve remains unaltered, and then changing by the ordinary methods of variations the form of the curve taken throughout the new limits. By the change of limits only, U becomes U' , where U' is given by equation (i) of the preceding problem, because that equation will hold irrespectively of the form of V. Then if in U' we change y into y -\- ^y, and y into y'-\- Sy\ and subtract W, we shall have the exact value of [^U']y to which, however, we can onl}^ approximate. This approxi- mation, so far as U' is concerned, is effected as in equations (2) and (3) of the last problem, which also hold irrespectively of the form of V. If now we take the variation of W^ in the usual way, we shall have first the terms <^V^dx^ — SV^dx^, which are evidently of the second order and must be rejected unless we are developing [<^U'] to the second order, when they must be added to those involving dx^'^ and dx^"". Next we obtain d6^ or / 'dVdx, where c^^ is to be developed to the first or second order as required, and the terms of the first order transformed as in the case of fixed limits. Hence to the first order we shall have [d ^] = V, dx, - V, dx, + P'd Vdx, which equation would evidently hold irrespectively of the form of V. But as in the present case F contains jk andy only, if we put as usual N for -— and P ior — -, and then develop dF to ay dy ^ the first order, and transform as usual, we shall have [SU^ := V, dx,~ V, dx.+Pfy-Pfy^^^'^' I ^V - g [ Sydx. (I) 92 CALCULUS OF VARIATIONS, 76. Now it will be remembered that the relation expressed in either equation (lo) of the preceding problem, not having been established upon any particular supposition, is true what- ever be the equations of the limiting curves. In this case, therefore, if we assume j —f{x) and y — F{x) to be the equa- tions of the two given curves, we can eliminate dy^ and dy^^ the terms of the second order which result from the elimina- tion being added to those already existing, or else being re- jected if terms of the second order are not to be considered. When these terms are to be neglected, equations (lo) are better written ^y. = (/-/). ^^. ^lo = {F'-y%dx,. (2) Performing this elimination, we have [cJt/] =. (^v+P/'-P/\d.r- {V+PF'-P/ldx„ Now having equated the terms of the first order to zero, it will appear that, as the integral cannot be made to depend upon terms which relate solely to its limits without in some manner restricting the generality of the function dy, we can only satisfy the equation ldU']=o by equating the integral and the terms at the limits separately to zero. It will be seen from (3) that [^ U] and (5^^ differ only in the terms at the limits, the integrals being identical, and this would be the case if F were any function whatever of ^, y, y\y", etc. Hence if we make the integral in (3) vanish, it must lead to the same general solution as though we had been discussing the problem of the brachistochrone between fixed points, and therefore the general solution must be a cycloid. It is clear, also, that if dx^ and dji\ be entirely independent, as they are in this case, we can only make the terms at the limits BRACHISTOCHRONE BETWEEN TWO CURVES. 93 certainly vanish by equating severally to zero the coefficients of these quantities. Performing this operation, and substitut- ing the values of V and P, we obtain for the upper limit x/:^zi_+i^i±z:Uo, whence by reduction we have and in like manner^ at the lower limit, we find equations which show that the cycloid must cut each of the two given curves at right angles. 77. We see, then, from the preceding examples, that if we wish to determine the conditions which will maximize or minimize any single definite integral in which the limits also are to be subject to an indefinitely small change, we have merely to put the integral, if possible, under the form U^^ I ^Vdx, F being some function of x, y, /, y\ etc., and then, if the general solution be known in the case in which the limits are fixed, we need only consider the terms at the limits, as the general solution will in every case be the same, whether the limits be fixed or variable. Moreover, if we wish to consider the terms of the first order only, the terms at the limits in \SU^ =0 Avill be identical with those which occur in SU —o, with the addition of the terms V^dx^— V^dx^. Then if no restriction be imposed upon the quantities dx^, dx^, Sy^, Sy^^ the coefficients of these quantities must be equated severall}^ to zero. This would give us, in addition to the usual 2n con- ditions, Fj^r o and Fo == o, equations which, as we have already seen, could not in general be satisfied, as we would have 94 CALCULUS OF VARLATLONS. 2n-\- 2 equations and only 2n arbitrary constants. But when the extremities of the required curve are restricted to two given curves, we can eliminate Sy^ and ^y^ as already shown, and thus the number of ancillary equations is reduced once more to 2n. Problem XL 78, It is required to determine the conditions which must hold at the limits^ when in Prob. III. we also demand that the required curve shall have its extremities upon two given curves. Assume, as before, the differential equations of the curves to be dy =f'dx,dy := F'dx. Then, following the last ar- ticle, we neglect all terms except those at the limits, since the general solution is known to be a cycloid. Here (r\ .i;^2\2 V=- — ~^ -— , and the terms at the limits, as will be seen y from Art. 30, will, after adding V^ dx^ — V^ dx^, become _ / 4/(i +/' ) , A ('+/T\ s^ But from equation (11), Art. 30, we have 4/(1+/") I ^ (i+ZT ^ , /' '^ dx f Moreover, we shall assume that the cycloid has cusps at the points whose co-ordinates are suffixed, in consequence of which * PROB. VII. WITH VARIABLE LIMITS. 95 y" will become infinite, and the terms in (i) which are divided \yy y" will vanish. Hence (i) becomes L=- c{py, - Sy,) = o. (2) But ^y^ = {f' — y')xdx^, ^y^ — {F' — y')^dx^, and substituting these values in (2), and equating severally to zero the coeffi- cients of dx^ and dx^^ we have f:-y: = o, F:-y:=o. But yl and y^ are equal, because the tangents to the cycloids at its cusps are parallel, and therefore the quantities jr/, y^, //, F^ are equal. Hence we conclude that the chord joining the two cusps of the cycloid must be normal to each of the given curves. Problem XII. 79. // is required that the generating ctirve in Prob. VII . shall have its extremities upon two given curves. Let the equations of the given curves be as in the preced- ing problem. Then V = y Vi +y^ and the terms at the limits become L = {yVi +y\ dx- (y Vi +y\ dx, _|_ (^Z_\ sy^ _ / /'^' \ dy^ = a (I) Eliminating Sy^ and fy^ as before, we have, after equating to zero the coefficients of dx^ and dx^. 9^ CALCULUS OF VARLATIONS, Whence reducing, we have which show that the catenary must cut its limiting curves at right angles. • Section IV. CASE IN WHICH SOME OF THE LIMITING VALUES OF X, Y, V, ETC., ENTER THE GENERAL FORM OF V. Problem XIII. 80, // is required to determine the nature and position of the curve down which a particle will descend in a minimum time from one given curve to another, all the curves being in the same vertical plane, and the motion of the particle beginni?ig at the point of its departure from the upper curve. Assume the axis of y vertically downward, and let x^, y^, x^, j/j be the respective co-ordinates of the initial and terminal points of the motion, and let the differential equations of the respective curves be ^ = F'dx, and dy =/V;r. Now in this case the velocity of the particle at any point whose ordinate is y will be \^2g{^y — y^, because the motion begins at the point whose ordinate is j,. Therefore in this problem we must minimize the expression U^r^±£^dx=rVd:c. (I) t/xo y y — y t/^o Although it at once appears that the limits x^ and x^ will also be subject to change in this problem, we see that one of these limiting co-ordinates, jf,,, enters likewise as a component part of F throughout the integral, and this fact will require BRACHISTOCHRONE CONTINUED. 97 some modification of our previous method of solution, because, since y^ is a component of F, any change in the value of /„ will produce a change in that of V throughout the entire range of integration. Moreover, the co-ordinates at the lower limit must always satisfy the equation y^ = F(x^y so that when we change x\ into x^ -\- dx^, we necessarily change y^ into F{x^ -|- dx^. It happens that V is not affected by any change in the values of the other limiting co-ordinates, as they do not occur in V\ but if they did, the method of treatment .would be similar to that which we are about to explain for y^. Now let V be what V becomes when we change y^ into J^o + '^o) ai^d we shall have, from the change of limits only, U'= / V'dx, (2) If we next change j/ into j/ + ^J> ^^d y' intoy-^ Sy\ and sub- tract U or J Vdx, the result will be the exact variation of U, to which we will now approximate as far as terms of the first order only. As before, to the first order, (2) becomes U '^ V/dx- V:dx,+ r V'dx, (3) Now when we change x^ into x^ + dx^, we to the first order change y^ into y^ -(- F^dx^, and therefore V is what V becomes when we change y^ mto y^ + F^dx^, y and y' in V being re- garded as constant, since they in no manner depend upon y^ ; and this change in V will evidently be -— F^dx^, where F^dx, dy, has simply been put for dy^. Hence to the first order, ^'=^+^^.''^-.- (4) Substituting this value in (3), rejecting again all terms of the 98 CALCULUS OF VARLATIONS. second order, and observing that F^' and dx^ may be regarded as constants, we have U'^ K dx,- K dx,+ F:dxX'^- dx +y"' Vdx. (5) If now we vary y and y' , we shall obtain the variation of / Vdx or U in the usual manner for fixed limits, while the variations of all the other terms must be neglected, being of an order higher than the first. Hence putting N for — -, Pfor -— , we shall have, after the usual transformation of SU, ay [^( I +y^) = ^ of Art. 25, by putting j—jTo for j. ^U — 7o) (i +/') ^^a ^ V2r. lOO CALCULUS OF VARLATIONS. Substituting this value in (lo) and (ii) after having restored the values of V and P, they become after reduction equations which show that the cycloid cuts the lower curve at right angles, while, since // = /^/, the tangents of the two given curves at the initial and terminal points of the motion must be parallel. 81. We have seen that when a particle starts from a state of rest, the cycloid must have a cusp at that point. But if it is to start with a given initial velocity in the direction of the tangent, which velocity could always have been produced by faUing from some height //, Fin Prob. II., Case 2, would be- come \^y + // If, as usual, we obtain the differential equation dP ax we can evidently, while keepings vertical, remove the axis of X to the height h above the initial point, without affecting the form of the curve given by the equation M — o. But making this change, y -{- h will become jk, and J/ will become identical with M in Prob. II., Case 2, thus giving us a cycloid with its cusps upon the new axis of x. That is, when the particle starts with a given tangential velocity, the curve of quickest descent, or the brachistochrone, will still be a cycloid, but having its cusps upon the horizontal, from which the particle must have fallen in order to acquire the given initial velocity upon reaching the starting-point. In like manner, in the last problem, if we require the par- BRACHISTOCHRONE CONTINUED. lOI tide to start from the upper curve with a fixed tangential velocity, due to some height h, V will merely become and no change will be effected in the results of the last article, except that the cycloid will no longer have a cusp upon the upper curve, but its cusps will then be upon the horizontal whose distance above the upper intersection is /i. 82. As examples of the kind discussed in the preceding problem are not numerous, we shall, as a means of more fully developing the method therein explained, now examine the terms of the second order. For greater simplicity, change .the independent variable, assuming the axis of x vertically downward ; and for greater generality, suppose the particle to start from the upper curve with an initial tangential velocity due to the height /i. Also let the equations of the curves he f = ^ W = ^ for the upper, and f — f{x) — fior the lower, while x^, y^, ^^, f,, are the co- ordinates of the initial and terminal points of the motion. Now we shall have Let F' be at once what V becomes when / is changed into y + ^/y and X, into x^ -\- dx,. Then we have ' [^^] =X+<.. ^'^- -X ^^-' (0 which is exact ; and we will now approximate to the second order. We have 102 CALCULUS OF VARLATLONS. J.., dV , dV r For brevity, let A denote all the terms of the first order ex- cept F, B those of the second, and C their sum. Then (i) becomes px-i^ + dxx f*Xy pxi + dxi ['^^]=X + ... Vd.-l^ Vd.^l^^^^ Cdx. (3) But, as formerly, the first integral in (3) gives F.^..-F,^..+ l{f)_^./-l(g')^^.; + XV^.. (4) Moreover, neglecting terms of an order higher than the second, the last integral in (3) becomes nx^ + dx^ nxi l+...^d.-^lBd.. (5) Also to the second order nxi + dx^ pxi / ,^ Adx = A,dx^-\-A,dx,-\- Adx. (6) JxQ-\-dxQ ^ ^ ' ° ° ' t/Xo ^ ' Hence, finally, we have + A,dx,-A,dx,-\-£ydx. (7) Restoring the value of A, transforming by integration, as BRACHISTOCHRONE CONTINUED. IO3 usual, the term / '-^, Sy'dx, and then eUminating dy^ and Sy^ *^^o ay by equations (10), Art 69, we have '+(f)y'-^).''--(f).('---'V^.+^.rj^'" +£'BJ.. (8) Making the terms of the first order vanish, we shall, as before, obtain the cycloid as the general solution, and it will be sub- ject to the conditions already explained. Then [^f/] will consist of the terms of the second order only, which must become positive if the solution give a true minimum. As the terms in Sy cancel, we shall have V +£'^^^- (9) 104 CALCULUS OF VARIATIONS. Now we cannot render it evident that this value of \pU'\ is necessarily positive, nor will any of our subsequent investiga- tions afford us the required assistance, there being no known method apphcable. Therefore, although the great Legendre erroneously supposed that we were sure of a minimum, we cannot in fact be certain of its existence in every case. (See Todhunter's History of Variations, Arts. 202, 300.) 83. When V contains several of the quantities x^, jKo, y^, x^, y^, j/, etc., the expression for [_SU'] becomes somewhat complicated. But as we know that to the first order the change which any function undergoes from an indefinitely small alteration in any of its components may be found by considering each change separately and then taking their sum, we may, as Prof. Jellett has suggested, use this method with advantage here, as we shall not require the terms of the second order. Suppose, then, that we have to maximize or minimize the expression U — j Vdx, where F is a function of x, y, y', and also some of the limiting values of these quantities, x^ and x^ being subject to change into x^ -\- dx^ and x^ + d^\- Fi'on^ the change in x^ alone, supposing the other quantities could re- main unaltered, U will be increased by V^ dx^ — V^ dx^. From varying y, /, etc., V would be increased by -1- ^y ~\~ ~ri ^y' H~ etc., or 6V, and U would therefore be increased by J^ SVdx. Lastly, by the alteration in the limiting quantities which enter it, F would, throughout the entire range of integration, be in- creased by ^ dx, + ^(3j/, -f 4^ (5>/ + etc., and the same for •^ dx, dy^ dy^ the lower limit. Calling this change S' V, U is increased by J d' Vdx. Adding these results, we have MIXED INTEGRALS. 105 \SU^ = V,dx, - VJx, + £y'Vdx +£yVdx = o. (I) Now the last integral in (i), being transformed as usual, will give us, besides certain additional terms at the limits, a differential equation M = o, and this equation will be the same in form as though V had not contained any of the limit- ing components. Hence the general solution will be the same as though Khad not contained these quantities, and the limits also had been fixed. Then, by using this general solu- tion, we must if possible, by definite integration, express the remaining integral in terms of suffixed quantities, our power to complete the solution being dependent upon our ability to remove this integral sign. After this has been done, we dis- cuss the resulting limiting equations as we would in any other case. Section V. CASE IN WHICH U IS A MIXED EXPRESSION; THAT IS, CON- TAINS AN INTEGRAI, TOGETHER WITH TERMS FREE FROM THE INTEGRAL SIGN. Problem XIV. S^. // is required to maximize or minimize the expression U = /"""y ^dx= r^ Vdx, Here Fis a function of y, /, y\ whence, by formula (A), Art. 56, v=c + py+Qy-y^, (I) and I06 CALCULUS OF VARIATIONS. Hence (i) gives and, by integration, y^^cx^d. (3) Now the terms at the upper Hmit are (^-fh + a^^'- and similar terms at the lower limit, so that unless some re- striction be imposed upon the independence of Sy^^ 6y^, Sy^\ and SyJ, there will be four limiting equations to satisfy, while the general solution contains but two arbitrary constants, and this will in general be impossible. But the above example, containing the first power only of y", the highest differential coefficient in F, is, as will be re- membered, a case of Exception 2, Art. 51. . It will also be remembered that it was shown by Euler's method, equation (8), Art. 52, that all such integrals can be reduced to a lower order, the expression taking the form W^ — W^-\- J V'dx^ a class of problems not yet considered. In the present case, recollecting that y"dx = dy\ y'dx = dy, we easily obtain, by parts. jT'^y ^~ dx = y^^ly; - y.^ly: - f\y^ - ^y'ly'dx = W,-W,+ r'V'dx, (4) Now if we vary y^, y/, we shall increase W^ by ^ tfr. + ^ *r/, or {nr-' iy'Sy\ +(^ sy) , MIXED INTEGRALS. IO7 and we can change W^ in no other way. A similar equation of course holds for W^. But these terms, relating to the limits only, can have nothing to do with the form of any general solution, which must, therefore, depend solely upon the form of V\ Now V is a function oi y andy only, and P= — ny'^ -^ly' — ny^ - ^ Hence by formula (C), Art. 56, we have, as before, ^yn - \y' =. C, y'*^ z=z CX-\- d. Now the terms at the limits resulting from the variation of V^ are P^^y^ — P^^y^, which must be added to those obtained by varying W^ and W^. Performing this operation, these terms become - {nyn-^\Sy,^{ny—\dy,+(^) dy,' -(^) (5>;. But these terms are the same as those which we obtained by discussing the problem as originally given ; and as the general solution is also the same, the difficulty which formerly oc- curred is not removed. 85. We may, however, from this example see how to pro- ceed in more important cases of mixed integrals which will hereafter occur. Thus, suppose we have to maximize or min- imize the expression U=W,-W,+ r^Vdx, vxo where W, and M^, are any functions of x„ y,, y/, etc., and x,, Jo> fJ'-'- y"'^ and V is any function oi x, y, / . . . . jj/H while the limiting values of x are also variable. As before, I08 CALCULUS OF VARIATIONS. if we change x^ into x^ -\- dx^ and vary y^, j//, etc., W^ will re- ceive the increment dW, , . dW,. . dW, . , . , -dv/^'^ + 'dy;''^-^^''^-^''''^ and VV^ will be increased in a similar manner by changing x^ into x^ -f- dx^ and varying j^, j^/o^ etc. These terms, being all suffixed, cannot control the general solution, which must be obtained by varying V in the usual manner, transforming the variation as previously explained, and solving the differential equation J/ = o which will be obtained. Then we have as the terms at the limits those derived from the transformation of dV, together with those derived from varying W^ and W^. Now if the limits be fixed we shall generally, in order that the number of limiting equations may not exceed that of the con- stants in the general solution, require that m shall not exceed n— I, the difficulty in the last problem arising from the fact that m is equal to n. But if the limits be not fixed, we shall also, as before, require usually some restriction upon the ex- tremities of the curve given by the general solution. Section VI. relative maxima and minima. Problem XV. 86. It is required to find among all plane curves of a given length which can be drawn between two fixed points, that wJiich, together with the ordinates of its extremities and the axis of x, shall contain a maximum area. Whatever be the nature of the required curve, we know that its length is y^ \/ 1 ^y^dx; and since it is to be com- PLANE CURVE OF MAXIMUM AREA. IO9 pared with curves of the same length only, its derived curves must not differ from it in length, and we must therefore have I d S^i-^ydx — o. But the enclosed area is / ydA;;and since this is to be a maximum for all changes in the form of the curve which permit its length to remain unaltered, we must have also to the first order / dy dx = o. Now in the problems hitherto considered no restriction has been imposed upon the variations of j/, y, etc., except that they must always be infinitesimal, and the curves given by the general solution have therefore been compared with all others which can be derived from them by such variations. The results, therefore, being subject to no restriction so far as the variations are concerned, are termed absolute maxima and minima, observing that the terms maxima and minima are used in their technical sense only, and not in that of greatest or least. But in the present problem we are to compare the required curve with such only as can be derived by infini- tesimal variations of y' without any increase in its length, and the area is to be a maximum with respect to such varia- tions only. That is, if we vary the required curve so as to increase its length, the area need no longer be a maximum. Examples of this nature, therefore, are termed problems of relative jnaxima and ijiiniina, and also isoperimetrical problems, and constitute the most numerous and important class of ques- tions discussed in the calculus of variations. 87. Resuming the equations of the last article, and treat- ing the first as usual, recollecting that Sy^ and Sy^ are zero, we have £yydx^O, (2) no CALCULUS OF VARIATLONS. which signify merely that any values of Sy which will satisfy (i) must also satisfy (2), it being supposed that the derived curve has been obtained. But although we are permitted to pass from the required curve to such derived curves only as do not differ from it in length, the number of such curves may nevertheless be infinite, so that we cannot express in an ex- plicit form the nature of the restriction which has been im- posed upon 6y, or rather upon Sy' ^ although we know that such variations could be given to y' as would not satisfy equa- tion (i), and might or might not satisfy (2). This restriction prevents us from employing our former reasoning, which d y' would here 2:ive the equations = o, the differen- ^ ^ dx 4/1 _^y2 tial equation of the right line, as appears from Prob. I., and d y the impossible equation i = o. Now put Z for dx 4/1 _py^ Then if Z can be a constant, it is evident that any values of Sy or dy' which will satisfy one of the equations at the beginning of the last article will satisfy the other also ; and we will now show that this is the only condition which will insure that ^y cannot be so taken as to satisfy one equation and not the other. 88. Let x^, x^, x„ x^ be four particular values of x chosen as hereafter explained, and let s denote the value of the inte- gral J Sy dx when the limits are ;ir„ and x.. and / its value when the Hmits are x^ and ^.. Then supposing the required curve to be obtained, let us make Sy zero, except from x^ to x^, and from x, to x, ; that is, leave the required curve un- varied in form except between these limits. Also let us give to ^y an invariable sign from x^ to x^, and an invariable but contrary sign from x^ to x^. Then we shall have r'dydx = s + t. (3) PLANE CURVE OF MAXIMUM AREA. Ill Now although neither s nor t separately vanishes, we can so take ^y that their sum shall vanish, and thus (i) be satisfied. Next let q denote the value of the integral / Zdydx when the limits are x\ and x\, and r its value when the limits are x^ and x^. Now the four values of x may also be so taken that Z will be of invariable sign from x^ to x^, and also from x^ to ^., it being of no importance whether the signs be the same or not for these two intervals. We can now, with the values of Sy formerly chosen, secure that, unless ^ be a constant, q and r shall be numerically unequal, and consequently that their sum shall not vanish. But, as before, r'Z6ydx = q + r, (4) tyxo and hence, if Z be variable, we can, without violating the re- striction which has been put upon dy, give it such values as will satisfy equation (2) but not (i), which is contrary to the conditions of the question. 89. Now since Z is a constant, let it equal -. Then a aZ = I ; and restoring the value of Z, we may write d y' \ — a dx 4/1 _|_y = O, (5) an equation which involves the coefficients of ^y dx in both (i) and (2), and is necessarily true, being equivalent to 1 — 1=0. But it will be seen that this differential equation, which combines both conditions of the question, would also have been obtained if we had at first required to maximize or minimize the expression U=S^l\y-^a^iJry')dx, 112 CALCULUS OF VARIATLONS. the exti^eme co-ordinates being fixed, and dy or Sy' being subject to no restriction. Moreover, we shall presently show that all problems of this sort can be treated in a similar manner. Now integrating (5), we have and solving for y\ we have Va — \c — X) Whence, by integration, we obtain y-^d= \/a'-(c-x)\ (8) which shows, if we employ, as we have, the positive sign, that the required curve must be a circular arc, in which a must be numerically equal to the radius r. 90. Suppose now, as just suggested, we attempt to maxi- mize or minimize absolutely the expression Here F is a function of y and y' , and P = , so that y I +/" by formula (C), Art. 56, we have ny y + aVi+y' = c + Whence VT+Y' ' '" i^-yy PLANE CURVE OF MAXIMUM AREA. II3 which must be solved thus : Va' -{c- yf where we still use the positive sign. Integrating this equation, we have x + d=^Va'-{c-~yf, (9) which evidently has the same interpretation as before, except that c and ^need not be identical with c and d of the last article. 91, It will be seen that besides the two constants which arise from the integration of (5), which we may call AI — o, we have also a third constant, a. But now we also have, be- sides the two ordinary conditions given by assigning the values of J, and Jo, a third condition, that the length of the circular arc shall have an assigned value ; and these conditions are sufficient for the determination of the three constants. Consider first the constant a. We know that the length of the chord of the given arc is V{x, — x,y -\- {y^ — yo)\ and is, therefore, determined as soon as the limiting co-ordinates are given ; and since the length of the arc is assigned, if we find an expression for the length of any arc in terms of its chord and radius, and then substitute in that expression the known values of the chord and arc in question, we can, by solving for a, determine its value definitely. This expression is {x, - x,Y + {y^ - y^)-" = 4r' sin^ — , (lo) where s is the length of the given arc, and a is numerically equal to r, its sign being reserved for future discussion. After the determination of a, the other constants are readily found. For, from the general equation of the circle, we have y,+ d=± Va'-ic-xy 114 CALCULUS OF VARIATLONS. and a similar equation for the lower limit ; and from these two equations, when the sign of their second members has been agreed upon, c and d can evidently be expressed in terms of a and the given limiting co-ordinates. 92. We will now, before proceeding further, consider the general mode of treating problems of relative maxima and minima. Suppose, then, we require that / vdx shall be a maxi- mum or minimum, v being any function of x, y, y'. . . .y^\ while at the same time / v^dx is to remain always constant, v' being any other function of x, y, /,... y^^\ Then because vdx is to be a maximum or minimum, we shall have to the first order t/Xo Xl dvdx = o; (i) /*Xi and because / v'dx is to be always constant, we must have absolutely r^dv'dx = o. (2) Now suppose the variations of these integrals to be found, and transformed by integration in the usual manner. Then if we make Sy^, (5>„, d>/, etc., zero, we shall obtain, from (i) and (2) respectively, results of the form fySydx^O, (3) £v'Sydx^O. (4) PLANE CURVE OF MAXIMUM AREA. " II5 But Sy being restricted, as hitherto explained, we cannot say that V and V must separately vanish, but equations (3) and (4) will certainly be satisfied simultaneously if we can be sure that V is always equal to V multiplied by some constant ; V that is, that — - is a constant ; and we will now show that no other condition will satisfy these equations simultaneously. 93. Supposing- the required curve to have been obtained, choose, as before, four values of x such that neither Fnor V shall change its sign while x lies between x^ and x^ or between x^ and x^. Now, as previously, vary the form of the curve between these two intervals only, and make the sign of dy invariable for each interval separately, giving to it the same or contrary sign for these two intervals, according as that of Fis contrary or the same. Then, although / Vdydx does not vanish when taken throughout either interval separately, we can so vary y as to make the integral taken throughout the second equal to the same integral taken throughout the first, but with a contrary sign. But we have £ VSy dx = /;' VSy d.r +£'' V6y dx, (5) dy being zero for the rest of the curve. Therefore (2) would in this case be satisfied. Now put / for -— , then (4) will be- come lyVSydx = o,- (6) Sy being supposed taken as before. But unless / be a con- stant, we can certainly select the four values of x so that the two integrals m the second member of (6) shall be numeri- cally unequal, in which case their sum would not vanish and Il6 ' CALCULUS OF VARIATLONS. (6), or rather (4), would not be satisfied. Hence /must be a constant in order to the existence of a relative maximum or minimum, since then any values of Sy which will satisfy (3) will also satisfy (4), while otherwise it would be possible, even from among the restricted values of (^J/, to select such as would satisfy one of these equations and not the other. The preceding demonstration is due to Bertrand (see Tod- hunter's History of Variations, Art. 312, and also the seventh volume of Liouville's Mathematical Journal, 1842), and the author most heartily agrees with Bertrand in regarding the ordinary method of treating this subject as insufficient. Now write I V then V+aV = V- V=o. But this equation, which involves V and V\ and, being true under all circumstances, is evidently sufficient for the solu- tion of the problem, would have been obtained if we had been seeking to render U an absolute maximum or minimum, where U— {z>-\~az'')d.v, and thus Ave are enabled to substitute for the given problem a problem of absolute maximum or minimum, the general solution of which will be identical with that which we require. This method is due to the illustrious Euler, who first re- duced the treatment of this class of problems to a simple yet comprehensive rule. (See Jellett, Introduction, page xvii.) It is evidently immaterial Avhich of the quantities v and 7>' we select to be multiplied by a constant. For if we have F— <7:F' = o, then V^ -\- /^V— o, where /^ = —. Moreover, we a may also give the constant multiplier any form which may be convenient, as — a, 2a, etc., its value being ascertained subsequently. PLANE CURVE OF MAXIMUM AREA. II7 94. Resuming the consideration of Prob. XV., let us now examine the terms of the second order. Here a difficulty presents itself in the outset which must be surmounted before we can proceed. We find that the variation of the area is simply / dydx, there being no additional terms of the second order ; so that if we equate this variation to zero, it would seem that the area could undergo no change whatever when the curve is varied, and that consequently we could have neither a maximum nor a minimum. But the supposition that the terms of the first order must vanish is only necessary when there are terms of a higher order, it being sufficient, in a case like the present, to suppose that they are zero so far as the terms of the first order are concerned ; that is, they need not be zero as regards Sy^, Nevertheless, as we cannot determine the nature of these terms of the second order, should any exist, we shall be compelled to change our method of investigation. Suppose, then, that we had required the curve of mini- mum length which, together with its extreme ordinates and the axis of x^ shall enclose a given area. Here the general solution will evidently be the same as formerly. For pro- ceeding as in the first three articles of this section, we shall obtain equations identical with (i) and (2); and moreover, by the last article, we see that by Euler's method we are now merely to maximize or minimize the expression U=ll\^'^-\-y"-^by)dx, where b = -. But the enclosed area, instead of being a maxi- a mum, IS now to be constant, so that / dydx is absolutely zero ; while the length of the required curve, instead of being constant, is now to become a minimum. Il8 CALCULUS OF VARIATIONS. It should here also be noticed that while the length of the required curve was to be constant, equation (i), Art. 87, can be true to the first order only. For since the variation of the length contains terms of an order higher than the first, and the entire series is to vanish absolutely, it is clear that the term of the first order must equal the sum of the others, taken with a contrary sign. As the area gives us no term of the second order, we have only that obtained from the variation of the required curve, which is ^-0 2/(i+yy -^ ^^ and if we regard Vi +y' as positive, the length of the curve is evidently a minimum. It must, however, be remembered that Sy and dy' are restricted to such values only as will satisfy the equation / dydx — o. But since (i) is positive for all real values of Sy' , we only require that the term of the first order in the variation of the length of the curve shall com- 1 , .1 . • • J • d * y' pletely vanish to msure a mmimum ; and smce — -^== ^ ^ dx \/i _^ya is a constant, this condition is secured when we make J Sy dx vanish absolutely. It will be seen that equation (i) would have been obtained had we found, according to Euler's method, the terms of the second order in b being -. But, as before, the variations are not entirely unrestricted, since they can have such values only as will makey dj^.r vanish absolutely. PLANE CURVE OF MAXIMUM AREA. IIQ 95. Now let us, according to Euler's method, consider the problem as originally given. Then we shall have U^ll\y + aVTT7^)dx. (2) Here ^ Hence dx 4/1 +y^ dx ^ ^(i+y? ' '■' y (3) where the last member has the negative or positive sign ac- cording as the circular arc is convex or concave to the axis of X. Therefore a := = ^t r, the positive or negative sign being used according as the circular arc is convex or concave to the axis of x. Making the terms of the first order vanish, (2) will give SU= - P ^_ "" _Sy'^dx= ± - r^- ^ _J d/V^.(4) 2 ^^^ |/(i -\^yy 2 ^^^ 4/(1 ^y'y Hence if the arc be concave to the axis of x, the area is a maximum ; but if convex, the area becomes a minimum ; and these results are evidently as they should be. It must, however, be remembered that we have not as yet shown that the use of Euler's method, as far as the terms of the second order, must in this latter case give necessarily a trustworthy result, but merely that this result is one which is known to be true. 96. We may now extend the discussion of this problem, and also that of relative maxima and minima generally, to the case 120 CALCULUS OF VARIATIONS. in which the limiting values of x, /, y\ etc., are also subject to change. We have already seen that if we seek to maximize or minimize an integral of the form U == J Vdx, the general solution will be the same whether we suppose the limiting values of x, y, y, etc., to be fixed or not. Hence when V be- comes, as it will in the use of Euler's method, v -}- av\ the general solution, obtained under the supposition that the limits are variable, will be identical in form with that ob- tained by supposing those limits to be fixed. Now suppose we add to Prob. XV. the condition that the required curve shall always have its extremities upon two given curves; and suppose that the two required points had been found and con- nected by the required curve. Then, unless this curve were a circular arc, we could evidently, from our preceding dis- cussion, vary it so as to increase the required area without changing the extremities of the curve. The general solution must therefore, as formerly, be a circular arc, the only ques- tion being to determine the position of its extremities. The reader will be able at once to apply a similar mode of reasoning to any problem of relative maxima or minima which may present itself; and therefore, without taking space to gen- eralize the demonstration, we shall assume that the general solution of these problems is, like those of absolute maxima and minima, the same in form whether the limiting values of ■^7 y, y\ etc., be fixed or variable. Hence, from what has been said, we see that Euler's method may be employed whether the limits of integration be fixed or variable, the problem being treated in all respects like one of absolute maxima or minima. 97. Assume, then, in order to discuss the limits, U^£^\y±r\/Y^Vy')dx. (i) PLANE CURVE OF MAXIMUM AREA. 121 If we suppose first that x^ and x^ are fixed, while y^ and y^ are variable — that is, that the arc has its extremities upon the two right lines whose equations are x = x^ and x = x^ — the terms at the limits evidently become which equations signify that the tangent to the arc at each limit must be parallel to x^ which is clearly impossible. But if one of the limiting values of y be fixed, the tangent at the other limit can be drawn as described, and it must be so drawn. Now suppose that the limiting values of x are to be vari- able also. Then the terms at the limits will evidently give the equation (_,±,Vi+y^)_^,,±,j^_Z_.^tf, (2) with a similar equation at the lower limit. Let the extremities of the arc be confined to two curves whose equations are y — Fix) ^ F, y — f{x) = /. Then eliminating Sy^ by means of equations (2), Art. yS, (2) becomes, after omitting the com- mon factor dx^, / r rfy' \ and a similar equation for the lower limit. But since ds, any element of the arc, equals Vi -\- ydx, (3) may be written , sm n . = r, ± ^ ^ cos m + sm ;// ' cos n \ : (4) 122 CALCULUS OF VARLATIONS. where m is the angle which the tangent to the arc makes with the axis of x, and n the angle which any tangent to the upper limiting curve makes with that axis. Let t be the angle which the tangents to the arc and the limiting curve make with each other at the upper limit. Then, since t is numerically equal to n — m, we have cos /j — cos in^ cos n^ -\- sin m^ sin n^. (5) Hence, clearing fractions, (4) gives r cos /j =: y^ cos 71^, and we can establish an equation of a similar character for the lower limit. It must, however, be remembered that none of these results concerning variable limits can be confirmed as true maxima or minima without an examination of the terms of the second order, which examination would be impracticable. Problem XVI. 98, // is required to determine the form of the solid of revolu- tion which shall possess a given surface and a maximum volume, the generating curve being required to pass through two fixed points 071 the axis of revolution. Assume x as the axis of revolution. Then the volume to be a maximum is / ny'dx, while the given superficial area which must remain constant is / 27ry Vi -\- y'"^ dx. Hence, omitting the constant tt, we have, by Euler's method, to maxi- mize absolutely the expression MAXIMUM SOLID OF REVOLUTION, 1 23 Hence, after the usual transformation, we have dU + r^ \ 2v+2a Vi +y' - 2a-^ — J^^ \ dydx, (2) which equation is evidently true whether Sy^ and Sy^ vanish or not. Here, as Fis a function of r and / onlv, and P = — , we have by formula (C), Art. 56, But since the generating curve is to meet the axis of x, c must vanish, and we have 2 I 2^j/ / 2a \ O. (4) Whence, My be not always zero, we have , 2a Hence and which, by integration, gives x + b= ± V^a'-f, (8) the equation of the circle whose radius is, numerically at least, 124 CALCULUS OF VARIATIONS. 2a, and whose centre is on the axis of x, thus rendering the required soHd a sphere. 99. We are evidently prevented, by the nature of this problem, from supposing that y can ever become negative, and we may therefore use the positive sign only in equation (8). For if we were to regard y as negative throughout any interval, say from x^ to x^, we would have the corresponding zone of surface negative, because dx and Vi -\^ y'^ are taken positively, which would be absurd. Hence we see from (5) that 2a is necessarily negative ; and using its known numerical value, we have 2<^ == — r. 100. We have now two constants, r and b, to determine, since we were obliged to make c vanish before we could com- pletely integrate equation (3). But it will be observed that it would have been sufficient for a solution had we merely required the generating curve to meet the axis of x at some point, taking this point as one of the limits, say the lower, and then regarding the limits as variable. By this method we would obtain a sphere, as before ; and then if we impose the condition that both extremities of the generating curve shall be confined to the axis of x, as is most natural, we shall have a complete sphere. Hence, since the superficial area is given, r^ is at once determined by dividing the area by ^.n, and the distance x^ — x^ being necessarily equal to 2r, becomes also known ; so that when one limiting value of x is given, the other can be readily found. Now from (8) we see that b is merely the abscissa of the centre of the circle or the sphere, and equals x^ -\- r, or x^ — r. As soon, therefore, as one of the limiting values of x is given, all the required quantities can be determined ; but if neither x^ nor x^ be given, r only can be determined. (01. Thus far there would seem to be nothing peculiar or unsatisfactory about our solution ; but we come now to speak MAXIMUM SOLID OF REVOLUTION. 1 25 of a point which has occasioned considerable discussion among mathematicians, and which has led to an important extension of the calculus of variations. Suppose that, as in the original enunciation of the problem, we require that x\ and x^ shall have assigned values, or that the value of x^ — x^ shall be assigned. Then the diameter of the sphere must be x^ — x^, and the only value which the sur- face of such a sphere can have is 7r{x^ — x^y, so that, as we are no longer at liberty to select a value for the superficial area, the solution appears at first to fail. But it has now been made apparent that the general solution of any problem of maxima or minima in the calculus of variations is entirely independent of any conditions which may be required to hold at the limits, the limits having been supposed fixed in the earlier problems for the sake of simplicity only. Therefore no general solution can be said to fail so long as it is always possible to assume such limiting values of x, y, y\ etc., as will satisfy all the con- ditions of the question which are necessarily involved in the general solution. In the present case, if we require that the surface of the solid may be entirely generated by the revolving curve, these conditions are merely that the value of the superficial area may be assigned at pleasure, and that the generating curve shall have both extremities upon the axis of ;r, which condi- tions can, as we have seen, always be fulfilled by a sphere. Thus, since no restriction of the limits x^ and x^ is implied in the method by which the general solution was obtained, the apparent failure of the solution, when these limits are assigned, appears to arise from imposing too many conditions upon the question, some of which are incompatible, and for this the calculus of variations is evidently not responsible. It will be remembered that in Prob. VII. we obtained as a general solution a catenary, having its directrix upon the axis of X, and then subsequently showed that the two fixed points could easily be so taken that no such catenary could be drawn. 126 CALCULUS OF VARIATIONS. In like manner, in Prob. XV. we shall be unable to draw the required arc if the given line be shorter than the right line which joins the two fixed points, or longer than a semicircum- ference constructed upon this right line as a diameter. In the first of these problems the conditions can, without changing the limiting values of x, always be satisfied by assuming suit- able values for y^ and y^, and a similar remark will apply to the second problem unless the length of the given line be less than x^ — Xq, in which case some change will become necessary in the limiting values of x also. The only peculiarity, then, about the present problem would seem to be that, while in the former two we are permitted to make various but not all possible assumptions regarding the quantities x^ — x^ and y^ — y^y here but one supposition regard- ing these quantities can be made for a given superficial area, and thus, as the probability of failure when we attempt before- hand to assign the limits, and also the surface, is vastl}^ greater in this problem than in the other two, it more readily presents itself to our minds. But we are naturally led to inquire whether there may not be some other solution for this and similar problems in those cases in which the general solution cannot be made applica- ble. This question, which has received much attention of late, and has led to an important extension in the calculus of varia- tions, will be discussed in a subsequent section on discontinu- ous solutions. It will here be sufficient to say that such solu- tions do in many cases exist, and are generally composed of arcs of curves, or of right lines, or of some combination of both, and they are hence termed discontinuous solutions. (02 Now if we put for 2a its value — r, the general equa- tion given by the terms at the limits is ^^y_ryV^+y'\d.r,-(--M^)lSy o, (9) MAXIMUM SOLID OF REVOLUTION. 12'] together with a similar equation for the lower Hmit ; and these equations are evidently hke those of the preceding problem, except that they are multiplied by y, and — r only is used. If we suppose x^ and x^ to be fixed, and j, and y^ to be variable, (9) gives V^i+/7. ""' \^^+/-A o. Hence we may have y^ = o, y^ — o, thus giving an entire sphere, which is satisfactory if the surface Avill permit. If one limiting value oi y be also given, the solution can always be effected, it being the closed segment of a sphere, having a given base and height, r being determined by the equation ^._ 4f_ (10) ^ - s - TtFf s being the given surface, and R the radius of the base. Re- garding the other solution, j/ = o, j/;' = o, it may be remarked that but one of these equations can ever be true, and therefore the other limit must be fixed. Now suppose the extremities of the generating curve to be limited to two other curves, all the curves being supposed to revolve about x, which is the same thing as limiting the sphere to two surfaces of revolution. Then, since the terms at the limits in this and the preceding problem compare as we have just shown, it will appear, by methods precisely like those employed in Art. 97, that we shall have ry^ cos /, = y^ cos 7i^, (i i) together with a similar equation for the lower limit. Thus we have either ji = o and y^ = o, giving a complete sphere, or else the relation given in the last equation of Art. 97. To interpret this relation, let aj> be the upper limiting curve. 128 CALCULUS OF VARIATLONS. p the point of intersection with the arc whose centre is c^py the ordinate y^ of the limiting curve, and np the normal. Then cpn = /„ and npy ~ n^, and we have r _ cos n^ ji cos tl (12) and this equation can only be satisfied by making cos t^ unity, which shows that the tangent to the limiting curve at/ must be parallel to the axis of x ; that is, that j, must be either a maximum or .minimum ordinate. But if y^ should become equal to r, this relation would no longer be necessary, for then the lines cp and yp would coincide, the angles cpn and ypn become the same angle cpn^ and (12) becomes merely T COS ci)7t - = £_, which determines nothing regardins: the direction r cos cp7i of the normal or tangent to the limiting curve ; and hence in this case the ordinate jj/^ need be neither a maximum nor a mini- mum. 103. It must not, however, be assumed that all the results obtained in the last two articles will necessarily render the volume either a maximum or a minimum. For we have already seen that it is always necessary to appeal to the terms of the second order before the results obtained by making those of the first vanish can be interpreted. We have, more- over, also stated that the discussion of these limiting terms, when the general solution is known, is a problem of the differ- ential calculus rather than of the calculus of variations, and particularly so when the terms of the second order are to be considered. As a means of illustrating both these remarks, MAXIMUM SOLID OF REVOLUTION. 1 29 we shall consider only the case in which one limiting- value of y is zero, and take the liberty, as that work is now inaccessible to most readers, of copying the discussion from. Todhunter's History of Variations, p. 408. Let j^ be any ordinate of the limiting curve, // the height of the segment, v its volume, and s its surface. Then, since the segment is known from the general solution obtained from variations to be always spherical in form, and by supposition has but one base, we have, r being the radius of the sphere. (-■-?)■ (0 7t \rh and we can now, by the differential calculus, determine the conditions which will render v a maximum or a minimum, sup- posing s to remain constant. Since s = 2nrJi is to remain con- stant, rh is a constant, say k^. Then from the equation of the circle, when the origin is at the extremity of any diameter, we have / = 2rh - h' = 2k'' - le ; whence je = 2k'-y\ and therefore (i) becomes . = n\k^S/W^f-^(WEir^ (2) Whence and since the differential of the limiting curve must be dy =y^dx, we have dv_ _ 7ry/(k'--f) . . dx ^2k'' - f ' 130 CALCULUS OF VARLATLONS. To make the second member of this equation vanish, we must have y ^ o, y =^ k, or y =. o. To test these solutions, write u = k^ — y^, z — V2k'' — y . Then dx J = J {^'^^r + ^W - 2zyy'^ + uyY^. (5) Whence it readily appears that if/ vanish, makings a maxi- mum or a minimum ordinate according as y" is negative or positive, V will have the like or contrary property to y ac- cording as u is positive or negative. \i y— k, without making y' vanish — that is, without being at the same time a maximum or aminimum ordinate — will be negative, and 7/ will be a maximum. But if y, while equal to k, be also a maximum or a minimum ordinate — that is, make y vanish — — -- will also vanish, and it will be found by trial 74 that the third differential will do so likewise, while — -^ will dx ' become negative or positive according as jj/ is a maximum or a minimum, thus making v have the like maximum or minimum property with j. We have already seen that the question does not permit us to suppose that J can become negative, and hence the limit- ing curve must be such that when y is zero it shall be a minimum ordinate, which will cause y' to become zero also. 74 These suppositions will render — i positive, having reduced the preceding differential coefficients to zero. Therefore the supposition that j/ is zero renders v a minimum. The foregoing results, which have been verified by the author, appear to be correct, although they do not agree with those obtained by Prof. Jellett. (See his page 165.) MAXIMUM SOLID OF REVOLUTION. I3I We have not yet examined the terms of the second order in the general solution obtained by the calculus of variations in the problem, as originally given, but shall resume this point hereafter. 104. It will be remembered that we were unable to in- tegrate equation (3), Art. 98 (that is, the equation M = o), without supposing c to become zero. Nevertheless this dif- ferential equation has been shown to be that of a curve traced by the focus of some conic section as it is rolled along the axis of X, and the following outline of the demonstration is, with some difference of notation, given by Prof. Jellett on page 364, but the proof is due to Delaunay. Let r =f{v) =/ be the polar equation of any curve, the pole being assumed at pleasure; and when that curve is rolled along the axis of x, let y = F{x) = F hQ the equation of the curve traced by that pole. Then the following relations are not difficult to estabhsh : By means of these relations we are sometimes able, when the equation (differential or other) of one curve is known, to deter- mine that of the other ; and such is the case in the present instance. Now write equation (3), Art. 98, thus : h = {f^d)^V^\ry^^ (3) where b = — 2a and d——c. Then, from (i), we obtain rdv 32 CALCULUS OF VARLATLONS. Substituting in (3) the values of y and Vi +y^ from (2) and (4), we obtain dv = _^£^_. (5) r ybr — r^ — d The integral of this equation is known to give I b J b' I - = — 7 — r —75 — -7 cos z; r 2d 4d' d =z-^^^^°^^' (6) where A =: —. If now we assume, as the polar equation of the conic section, I _ I -\- c cos V r ~ A{i -e") ' ^^^ we can obtain from it equation (6) by merely making e equal / d' to y I — , and hence the truth of the proposition is estab- A lished. The curves which may be thus described are exceedingly various. Thus, if we make d =: — c vanish, the conic section will become a straight line, and the curve generated will be a circle, giving a sphere as a general solution, which agrees with what has been already shown. Moreover, the circle is evidently the only one of these curves which can ever meet the axis of x. Again, if we take the circle as the conic sec- tion, the curve, traced by its focus — that is, its centre — will be a right line parallel to the axis of x, and the required solid will be a cylinder. CURVE OF LOWEST CENTRE OF GRAVITY. 1 33 Problem XVII. 105, // is required to determine the form which a uniform cord of given lengthy whose extremities are confined to two fixed points or curves^ must assume in order that its centre of gravity may be at a maximum depth. Take the horizontal as the axis of x, and let L or be the given length of the cord, which, by the conditions of the problem, is to remain constant. Then, by the well-known principles of finding the co-ordinates of the centre of gravity of any curve, we shall have, D being the depth, which is to become a maximum, Hence, by Euler's method, we are to maximize absolutely the, expression Here Fis a function of j/ and y' only, and p= yy' -^ "y' , d) SO that by formula (C), Art. 56, we have 134 CALCULUS OF VARIATIONS. Whence, by reduction, we obtain y -|- aL and - cL, (3) which, to be rendered integrable, must be solved thus : Integrating this equation, we obtain x^ Al{y-\- B ^ ^{y ^ By - A") + C, (6) where A — cL, B — aL. Comparing this equation with equa- tion (5), Art. 59, we see that it is also the equation of a cate- nary, in which y -^ B \^ put for y ; because the reasoning in Art. 59 will apply equally to any curve whose equation is of that form, and this equation will take that form if, while keeping the axis of x horizontal, we remove it so as to make B zero. Indeed, without integrating, we may at once reach this conclusion. For by comparing equation (3) with equa- tion (2), Art. 59, we see it- to be the differential equation of a catenary, as described. 106. To determine the constants A^B, C, we have the con- ditions that the curve must, if its extremities be fixed, pass through those fixed points, and must have also a given length, and these three conditions are sufficient, assuming that we can solve any exponential equation which may arise. Com- paring (6) with equation (5), Art. 59, we see that if we make the axis of x pass through one of the given points, and esti- mate y upward, B will be the distance from the axis of x to CURVE OF LOWEST CENTRE OF GRAVITY. 1 35 the directrix, estimated positively ; but if we estimate y down- ward, B will have the same numerical value, but will be nega- tive. We adopt, however, the former supposition. Then, as L is positive, a ox — must be also taken positively. We may, if we choose, dispose of the constant C, as we did of the con- stant b in Art. 59, by making \t — AlA. If, then, we can determine A^ the discussion of the con- stants will be complete ; and this may be done in the follow- ing manner: Let D denote x^ — x^, which is supposed to be known, and £, j\ — jKo, which is also known, and let the ordi- nate, drawn to the lowest point of the catenary, divide D into two segments, / and ^, while the corresponding segments of the arc L are m and n, so that we have /+^-A ' (7) m-\-n = L. (8) Then, in discussions of the catenary, the following equation is easily established : 7n = ^{e'-e-^), (9) together with a similar equation between ^ and Jt. Whence /J / L -L L -i\ L=-le^-e ^J^e^-e ^V (10) Now because the catenary passes through the two fixed points, we have from its equation, (10) of Art. 59, A / L -I ^ -^\ (II) 136 CALCULUS OF VARLATIONS. which equation, combined with the preceding four, will evi- dently determine A, which in statics denotes numerically the tension which the cord will sustain at its lowest point. 107. If the extremities be not fixed, but merely confined to two curves, the general solution will of course be un- changed, only certain conditions must hold at the limits. For now the limiting terms, which vanished when the extremities were fixed, become V^dx,- V,dx, + P,dy,-P,6y, = o. (12) Substituting the values of P and V from (i) and the preced- ing equation, (12) gives, for the upper limit, together with a similar equation for the lower limit. Let the equation of the upper limiting curve be dj/=/'dx. Then eliminating (5>, by the equation (13) gives its numerical value in either case being the same, and a maximum. 109. If we assume the vertical as the independent vari- able, the general solution must be the same whether we can obtain it by that method or not. For whatever change can be made in the form of the required curve by ascribing varia- tions to y and its differential coefficients with respect to x^ can, at least if the curve be continuous and drawn between fixed points, also be made by ascribing suitable variations to x and its differential coefficients with respect to y, y itself re- ceiving no variation. This principle will be found to aid us in the solution of the following problem. Problem XVIIJ.* 110. It is required to draw between two fixed points A and B a curve of given length having the following property : that if at any point S of the required curve an ordinate NS be drawn, and on it we lay off NP equal to the arc A S, the curve traced by the point P shall enclose a maxiinum or minimum area. N * This problem is only a particular case of the second of the celebrated iso- perimetrical problems given by James Bernoulli, the original problem requiring NP to be any function of the arc AS, which can, of course, not be fully solved so long as the nature of the function is entirely undetermined. The solution is from the Adams Essay, Chapter XI. JAMES BERNOULLFS PROBLEM. 139 Here the area to be made a maximum or a minimum is I ^sdxy s being the length of the arc measured from A, while / ' |/i j^ ydx (that is, the length of the arc A SB) is to re- main constant. Hence we are, by Euler's method, to maxi- mize or minimize the expression u=Z\^+^^^+yY-- ■ (I) Hence, to the second order, su=r'\6s-\- /-^'^ _ dyH — —^ sy' \ dx. (2) But ^=£y'+y'^^' Whence ds ^ r ^ ^ dy dx + - r J^ — dy dx, (3) and / c^^ ^,r = xds -f X -£ dx. (4) Hence, taking this integral from x^ to x^, and observing from the figure that x^ is zero, while b may be put for x^, because it is constant, we have rssdx^b r\--jL==-6y4 ^ ^ Syldx "^^ ^" ( Vi+y -^ 2i/(i +yy ) - f\\ y' - dy-i ^ - dy^l^;r ^0 ^ |/i_|_y^ -^ ^2i/(i +y7 S = f\t-x)\ , ^' sy + ^^1_ dy^ X dx. (5) 140 CALCULUS OF VARLATIONS. Substituting this value in (2), and employing the usual nota- tion for the limits, we have dU= r\a -^b- x)\- A Sy' -\ -1 Sy'^ X dx. (6) Now examining the second factor of the second member of (6), we see that it is the variation of Vi +y dx, or ds, and that b — X, or Z, is the distance of any point of the arc ASB from the line BF, and therefore it is not difficult to see that the prob- lem is really in solution as though, taking the vertical as the independent variable, we had required the form of the curve of given length passing through A and B, and having the dis- tance of its centre of gravity from BF a maximum or a mini- mum. Therefore, without solving (6) in detail, we know from the last article of the preceding problem that this curve is a catenary, having its directrix parallel to the axis of y. in. But some investigation will be necessary in order to determine the sign of the terms of the second order. For although, as before, it is evident that a, like B of the last prob- lem, is numerically equal to the perpendicular distance from BF to the directrix, its sign is not at once clear. Treating the terms of the first order in (6) in the usual way, we obtain {a^b- x)--l=._^c, (7) Whence ,+,_. = ii:i+Z". (8) y Differentiating (8), and dividing by dx, we have cf ywi+y = h (9) SOLID OF MAXIMUM ATTRACTION. I4I from which it appears that c must always be of the same sign as y" . But the catenary may be either convex or concave to the axis of x, so that c will be positive in the former and negative in the latter case. Moreover, we see from (7) that a-\-b — X must always be of the same sign as c, and therefore the terms of the second order will become positive when the catenary is convex to the axis of x, and negative when it is concave, thus showing that the area in question will be a minimum in the former and a maximum in the latter case. Problem XIX. 112. // is required to determine the form of the solid of revo- lution of given mass and iinifor^n density which ivill exert a maxi- mum attractive force upon a particle situated upon the axis of revolution. Take the axis of revolution as that of x, and let the at- tracted particle be at the origin. Moreover, conceive the solid to be divided into shces of the thickness dx, by planes perpendicular to x. Then, by dividing these sHces into dif- ferential rings, it is easily found that, omitting the factor of density, because constant, the force exerted by any slice in the direction of x is 27r ; I , ^ \ dx. Hence is to be a maximum, while the volume / ny'^dx is to remain 142 CALCULUS OF VARIATIONS. constant. Hence, by Euler's method, we maximize the ex- pression U= r'\i ^J. + a/\dx = r'Vdx. (i) Therefore, to the second order, we have 6U= r \ ^-^^-3 + 2ay \ Sydx .r(.r' — 2/) +rl"-^ 2(x^-\-yy Sfdx. (2) Here F is a function of x and y only, and the terms of the first order in (^^need no transformation; so that we have at once, unless y be always zero, ^ ^ + 2^==o. (3) {x^+/) Now putting for 2a, (3) gives (^^+y)i^,^^. (4) But V, the volume, or / rty'^dx, is a known quantity, and (4) gives y = (,V)§_^^ (5) Whence v=7t r\c^ x^ - x') dx, (6) But the general integral of the second member of (6) is ;r(l.J^S-ly)+«?. (7) SOLID OF MAXIMUM ATTRACTION. 143 Now suppose x^ to be zero, which will place the particle upon the surface of the solid ; and assume also that when ;r == jTj the generating curve meets the axis of x. Then, by making y zero, and x, x^, in (4), we see that x^ is numerically equal to c ; and taking (7) between the limits o and c, we find v = ^, (8) 15 It therefore appears that, when the volume is given, the length of the axis is not in our power, but is determined by that volume ; and c^ being known, a is also known. 113. Now the coefficient of Sy" dx in (2) is Putting for a its value ^, and substituting for the first members of (4) and (5) the second members of the same equa- tions, (9) becomes I xf^ix" - 2c^ x^) I 3.^-t - 2c% 2C'^ 2{ex)l ' 20"^ 2C^ ' Hence the terms of the second order become Now, since v cannot be negative, we see from (8) that c must be positive, and it is numerically greater than x^ being equal to x^. Therefore E is positive, while Z can never become positive, and the terms of the second order become invariably negative, thus giving us a maximum. 144 CALCULUS OF VARLATIONS. Problem XX.* 114. It is required to determine the form of the solid of revo- lution, having a given base and volume, which will experience a mijii^num resistance in passing through a fluid in the direction of the axis of revolution. Let X be the axis of revolution. Then, reasoning as in Prob. VI., we see that we must minimize / -^'-^ , dx, while, the volume being given, we must have / y^dx constant. Therefore, by Euler's method, we minimize absolutely the expression Here Fis a function oi y andy, and so that by formula (C), Art. 56, we have We will assume that the generating curve cuts the axis of X, which will render b zero, and then we easily obtain the equations -y = (T^= ^"d > = -^„ (4) c being put for -. a * The following discussion, which is much more satisfactory than that of Prob. VI., appears to be almost entirely due to Prof. Todhunter. (See his Adams Essay, Chapter X., from which this solution is taken.) PROB. VI. WITH GIVEN BASE AND VOLUME. I45 115. Now the last equation can be shown to indicate that the generating curve is a hypocycloid. For let y' = tan v. Then, by (4), we have y = c sin^ v cos v^ (5) and, by differentiation, we have y = c(^ sin^ V cos^ V — sin* v) — dx di) = ^(3 cos'' V — sin'' v) sin^ v -7-. (6) Hence dy -^ = ^(3 cos'' V — sin* V) sin* v, (7) Dividing (7) by y' = tan z' = ^-, we have cos V dx — - = ^(3 cos' V — sin* z)) sin v cos ^. (8) Squaring and adding (7) and (8), observing that sin* V -f- sin* v cos* v = sin* 7/ (sin* v -\- cos* ^) = sin* ^.i, we have, putting ds for an element of the arc, ds — = ^(3 cos* V — sin* v) sin ^ r=r ^ sin ^v. (9) To integrate, write this equation thus : c ds = — sin 3^/^(3^/). Then we obtain s= - ^cossv + c,. (10) 14^ CALCULUS OF VARIATIONS. This equation is known to indicate that the curve is a hypo- cycloid, the radius of the rolHng circle being one third that of the fixed circle. If now we suppose that when y van- ishes V vanishes also, and measure s from this point, we have o = cos o-\- c^\ that is, <:=: — ; and (lo) becomes 3 3 ^ = - (i — COS37/). (11) 116: To determine the constant c, we have the conditions that the solid must have a given base and an assigned volume, and we may use these conditions thus : Let v^ be what v be- comes when X ^=^ x^ and when j/ =:j|/j, a known constant, say B, Then it is shown that the volume of the sohd is TtB'i^- sin^ v,A sin' v\ , , V8 10 '^3 V (12) We have also, from (5), B ^^ c^\vl v^CQS>v^\ (13) and from these equations v^ and c may be determined. This solution, however, like some others, is not always possible. For it is shown that the volume can be as great as we please, but that it diminishes as v^ increases, and has its least value when v^ = — , its value then being — • If, therefore, the given volume be less than this quantity, no such solid, with the given volume, could be constructed upon the given base. 117. Let us now examine the terms of the second order. We may evidently write 1/ thus : U=£\.yf+2af)dx, (1-4) PROB. VI. WITH GIVEN BASE AND VOLUME. 1 47 where /=: — - — ^, and is therefore a function of y' only. ■^ 1 4-y' •' Hence the terms of the second order arising from the expres- sion / yfdx may be trea1:ed as in Prob. VI 1 1., while the term arising from / 2ay''dx is evidently - / \aSy''dx. Therefore, by the formula of Prob. VIII., we have, Avhen we suppose the limits to be fixed, where and # f,^^^ 3/1+/' ^ dy' (I ^-y'r (16) But, from the first equation (4), we have vH-iv" — 1/'*') V'= ^ (l^yy -' ^^hence 2a =/"/'. (18) Substituting this value in (15), we have ^^= \S,J\y^y''+y"^f) dx. (19) Now f" is positive so long as y does not exceed three ; that is, when v does not exceed -; and 3 that the integral becomes positive. is, when v does not exceed -; and y" is here positive also, so 148 CALCULUS OF VARIATIONS. 118. But since the distance x^ — x^ is not fixed, it is evident that the limits of integration are not altogether fixed. But as the base is given, and we may consider its distance from the origin as fixed, the Hmit x^ may be regarded as fixed, as is also j/j. Now the terms of the first and second order arising from the variation of x^ and }\ evidently are - r. dx, - Pfiy^ - i ^dx: - S V, dx, - L// Sy:, (20) the last term resulting from the formula in Prob. VIII. , when dy at either limit does not vanish. But P. = yJ^. -j-- = 7o /o' y" +/o Jo' + 4^jo/oi dx, ^K = yj: ^y: +/o ^y. + A^y. ^jv Now since y^ is zero, and, as appears from (4), j/ is also zero, all the quantities V^, f^, //, -— - will separately vanish, and the UXq terms in (20) will disappear. Therefore the variation arising from a change in x^ and y^ is not even of the second order, although it might still be a quantity of the third order; and as the integral in (19) is positive, we have in this case a solid of minimum resistance. Problem XXI. (19. Let a curve meet the axis of x at tzvo fixed points, the origin being assumed midway between them. Then it is required to determine the form of this curve, so that, being revolved about the axis of x, it may generate a solid of given volume whose moment of inertia, with respect to the axis of y, may be a mini- mum. SOLID OF MINIMUM MOMENT OF INERTIA. 1 49 Conceive the solid to be divided into slices by planes per- pendicular to the axis of x. Then the moment of inertia of any slice, whose thickness is dx, is 7tin I — '(-+xy\dx, (,) where m denotes the mass, which is constant. This equation is easily obtained from the moment of inertia of the rings of which the slice is composed, which is m'^+-A (2) J/ being the mass of the ring, or 27tmdydx. Therefore, since the volume is to remain constant, we must, by Euler's method, minimize the expression ^' -X? 1 f + -y - -y \ ^- = X? vdx. (3) Of course we could have put a for — a^ as the indeterminate multiplier, and this is what we would naturally do in first in- vestigating the problem ; still the present form is known to be more convenient. Now we have ^^= £yy^+ ^^'-^ - ^^^^-^ ^-^ =Xy^y^-^ 2'^'- ^"^yy ^^' (4) Hence, if y be not always zero, we have / + 2.r^==2^^ (5) which shows that the solid must be an oblate spheroid m which the major axis is to the minor as V2 is to unity. ISO CALCULUS OF VARIATIONS. 120. The terms of the second order are which, by means of (5), reduce to SU = / y'Sy'dx, and this being- necessarily positive, we have a minimum. But while the solution is thus apparently satisfactory, it evidently affords another example of the kind discussed in Prob. XVL For if we suppose the limits x^ and x^ to be assigned — that is, the minor axis of the ellipse — then, unless the volume be just , in which B is the semi-minor axis, no such spheroid can be constructed. But if, without assign- ing the limits except to make the curve meet the axis of x at two points equally distant from the origin, we only require the figure into which a given volume must be formed, as above, we shall obtain a spheroid in which the axes are related as just mentioned, the limiting values of x having been determined by the given volume. Still, in the investigation of the terms of the second order just given, we have assumed that x^ and x^ undergo no change. Nevertheless, if we vary x and y at the limits, we shall not increase these terms, since, y at the limits being zero, F^, V^,SV^,dV^A-j-\ and (-^j severally vanish. Here the constants are all determined by the assigned volume, combined with the conditions that y^ and y^ shall be zero. For B is determined from the condition that the volume must equal an assigned quantity; then A, the semi-major 3 axis, by the known relation between the axes ; after which a" is found by means of (5). PROBLEM OF LEAST ACTLON. I5I Section VII. CASE LN WHICH V IS A FUNCTION OF POLAR CO-ORDINATES AND THEIR DIFFERENTIAL COEFFICIENTS. 121. The principles of the calculus of variations thus far obtained are equally applicable when polar co-ordinates are to be employed ; and as the mode of applying these principles is precisely similar to that which we have already given for rectangular co-ordinates, we shall present but two examples, the first of absolute, and the second of relative maxima and minima. Problem XXII. A particle which is always attracted towards a fixed centre^ with a force which varies according to the Newtonian law of gravity^ is projected from a fixed point so as to pass through a second fixed point. It is required to determine the nature cf its pathy assuming that it must be the path of least or minimum action. Assume the attracting centre as the pole, r as the radius vector, or distance of the particle at any time, from the centre of force, r^ and r, the distance of the first and second points respectively, and v the natural angle included between r^ and any other radius vector. Also let /, a constant, be the inten- sity of the force at a unit's distance, and v' the velocity of the particle in its orbit at any instant. Now, from mechanics, the action of the path is t/So 'ds, (I) where ds is an element of the path. But ds = Vdr'+Vdzr' = dvV r' + $^' = Vr^+VV^, (2) dv 152 CALCULUS OF VARIATIONS. SO that the action becomes £\J .fTTj^dv. (3) Now in determining v' three cases arise. For we know that the path of a revolving particle will be an eUipse, a parab- ola, or an h3^perbola, according as v^, the velocity of projec- tion, is less, equal to, or greater than y — . Let us here con- sider the first case, and suppose v' — y — — --. Then it is r, a known that v' will equal /^~^. (4) r a Substituting this value of v' in (3), and omitting the constant /, we have to minimize absolutely the expression ^fyVT+T-'d.^fydv. (5) Now change r into r -\- Sr, and r' into r' -\- Sr', while v re- mains unvaried. Then we can develop the new state of U just as we could if in U we had put x for v, y for r, and y' for r' . Hence, to the first order, we have PROBLEM OF LEAST ACTION. 1 53 But, as in plane co-ordinates, dr' = --— , so that dU may be dv transformed in the usual manner by integration by parts, Sr^^ and Sr^ vanishing because the two radii are fixed. But we need not perform this transformation, which would give an expression not readily integrable. For the formulas of Art. 56 become at once applicable to polar co-ordinates when in those formulae we substitute v, r, r\ r\ etc., for x, y, y\y\ etc. Here, then, F is a function of r and r\ and dV Wr' P o^ -rr — — , (n\ dr 4/^2_j_^/2 v// so that by formula (C), Art. 56, we have Wr"" , , Wr' W Vr'-{- r"' = __!__ . + c, and '' ' = c, (8) Vr'-\- r' Vr'+ r'' Solving for r^', we obtain r = (9) where 3 = c\ Now put - for r. Then the following equa- tions will be found to hold true : W'=2U- -, a r'': I dtl' ti'dv"' and (9) gives dll^ _2U dv' b I -u\ (10) Solving and putting C for — , we have ab J du dv = b 154 CALCULUS OF VARIATIONS. where the negative sign is used, because -- = — . Now dv r dv by placing 75—75 within the radical sign in (11), that equa- tion may evidently be written thus : , — du dX dv = ^t-'^) -{«-,-)■ ""-"' ""' Whence, by integration, we obtain I I -1 ^ -\ b V -\- g— cos -7,- = cos K Whence and ^ b' I ^ — 7- COS(z/+^)r^ . (13) ^ b' u or i = ^+r ^— ^'cos (^ + ^). (14) Now write b = a{i — /), and replace C by its value --. Then c the quantitv under the radical readily reduces to —. --, ^ -^ -^ a{i — e") and we have I _ i+^cos(2;+^) Now in equation (8), in order that c, or Vb, may be a real quantity, we must, since a is by supposition positive, have PROBLEM OF LEAST ACTION. 155 I _ e" positive. That is, e must be less than unity, and (15) is therefore the equation of an ellipse. 122. It appears as though the general solution contained four arbitrary constants ; but as e depends upon the ratio of a and by the semi-major and minor axes, the number of arbitrary constants is only three. But, as in former examples, the gen- eral solution is totally independent of the possibility of render- ing it appHcable in any particular case. Of these constants, a, or the semi-major axis, is determined as soon as/, r^ and < are given, but must of course be of sufficient value to enable the ellipse to pass through the second fixed point. The least value of a which will render the solution possible in any case may be determined thus : Since the distance of the two fixed points from the first focus are respectively r^ and r^, their re- spective distances from the second focus must be 2a — r^ and 2a — r^. Now from the first fixed point, with a radius 2a — r^, and from the second, with a radius 2a — r^, describe circular arcs. Then if these arcs do not touch there can be no solu- tion, the least admissible value of a being that which will cause them to touch, while if a be increased beyond this value, the circles will cut, and there will be two positions for the second focus , that is, two ellipses can be described as required. Thus, although we seem to have three conditions for the determination of the three constants — namely, the intensity of the initial velocity and the distance of each of the two fixed points from the focus — we can in fact only determine a. This result might, however, have been anticipated, as we know from mechanics that while the form of the curve and the value of its major axis depend solely upon the values of /, v^' and r^, the minor axis, 2b, is also dependent upon the direc- tion of the initial velocity, the equation of condition being Wl sin m,, (16) 156 CALCULUS OF VARIATIONS. where m^ is the angle which the orbit at the point r^ makes with r^ ; and this element of direction we have thus far entirely ignored. If now we assign the value of m^, b and conse- quently e will be given by (16), and ^ must then be determmed by making the ellipse pass through the two fixed pomts. When a has its least value, so that but one ellipse can be described, the chord joining the two fixed points is evidently a focal chord ; and when a permits two ellipses to be de- scribed, one of them will have its foci upon opposite sides of this chord, while the other will have both upon the same side. This distinction is important, as we shall subsequently show by Jacobi's method that only when the ellipse is of the latter species does it render the action a minimum. 123. If, with a fixed. value of r^ and v^, we regard m^ as variable, and for each value of m^ cause the second fixed point B to assume the corresponding position, which would render one solution only possible, the point B will itself always be found upon the perimeter of an ellipse. For there being but one solution, if D be the chord joining the two fixed points, the circles described as above will just touch on D, and we shall have 2a-r,-\-2a — r,—D, or Z>+r, = 4^ — r„. But D and r, are variable, while a and r^ are constant. There- fore, since the point B is always so situated that the sum of its distances from the first fixed point and the centre of force is always equal to a constant, it is on an ellipse whose foci are at these two points, whose major axis is d,a — r^, and whose eccentricity is — ^- — ; and we may call this eUipse the limit- ing ellipse. (24. We may, in closing, advert to the two remaining cases of this problem., PROBLEM OF LEAST ACTLON. 157 Suppose, first, that we make v^ equal to r — . Then it is known that v' will equal y — ; and proceeding precisely as in the former case, or better by making C zero in equation (14), (since that equation is true when - is zero,) we shall obtain I _ I +^cos(^+^) r (17) the equation of a parabola, in which b is one half the latus rectum. Suppose, secondly, that we have v^ = y ^^-\-:L, Then = i/'^+C / 2f f we know that v' will always equal r — + -- ; and proceed- r a ing in all respects as before, we shall obtain, in the place of equation (14), ^ = ^ + r ^.+ <^cos(^+^), (18) where C has the same value as in (14). If now we write b=^ — a(\ — e^), (18) will readily reduce to £ ^ _ I ±ccos{v-\-g) r a(i-e') ' ^^^^ But we shall, in the course of the investigation, obtain an equation identical in form with (8), except that W will equal y — I — . Hence, that c or Vb may be real, b or — (i — e"") r a ^ ^ must be positive ; and therefore, since a is by supposition positive, it readily appears that i — ^' is negative ; so that 158 CALCULUS OF VARIATIONS. since e in this case is greater than unity, (19) becomes the equation of an hyperbola, having its attracting focus within the curve. This is as it should be, since a particle, revolving in an orbit according to the Newtonian law, can never de- scribe an hyperbolic arc having the attracting focus without the curve. Problem XXIII. 125. It is required to determine the form of the plane closed curve of given length which will envelop a maximiiin area. Assume the pole within the figure, and let / be the length of the given perimeter. Then, because the curve is to be closed, we have S/r^J^r'-'dv, (i) which is to remain constant. Now m being the enclosed area, 2 we have we have, by the principle of polar areas, dm = — dv, so that r^r'dv . , , which must become a maximum. Now the reasoning of Bertrand, in Arts. 92 and 93, is evi- dently rendered applicable to polar co-ordinates by substi- tuting V, r, /, etc., for Xy y, y, etc. Whence we conclude that Euler's method may be used for polar co-ordinates just as it has been hitherto employed. We must, then, maximize abso- lutely the expression U=J V- + ^V9^-\-r''\dv=l Vdv. (3) CLOSED CURVE OF MAXIMUM AREA, 159 Here F is a function of r and r' , and ar' ""-VTW^' ^'' so that by formula (C), Art. 56, we have 2 ^ ^ VT+T"^ ' and r'-4- ■ '^+ -^J^T^' =^'- (5) Therefore = 7^-20, (6) Whence ^ 4^V- _ . ^ . 4^V- _ (^ - 2.y ir'- 2cY {r' - 2cy Hence dv 7^ — 2C dr r V4a'r' - {r' - 2cf (7) (8) Now squaring r" — 2c under the radical sign, dividing both numerator and denominator by r^^ and then placing within the radical the quantity 4^ — 4^, (8) may be written thus : Write ^ = r + ^. (,o) i6o CALCULUS OF VARIATIONS. Then (9) becomes dv = dZ i/^a' + SC-Z' Therefore, by integration, we obtain Z and V -\-g=i sin" sin(t^+^) Z V4a'+Sc (II) (12) (13) Clearing fractions and restoring the value of Z, then clearing fractions again and transposing the first member, we obtain r^ — 2r Vd" -\- 2c sin {v ^ g) -\- 2c — o, (H) which is one form of the polar equation of the circle when the pole is assumed at pleasure, a being the radius. (26. Equation (14) is the form in which the result is left by Prof. Todhunter. (See his History of Variations, Art. 99.) To interpret this result, let P be the pole, APB a diameter, and denote PA by C. Then since the equation of the circle, when the origin is at-^, a being its. radius, is y = 2ax — 4^^ if we remove the origin to P, it will become f=2a{x^C)-{x-^C)\ (15) CLOSED CURVE OF MAXIMUM AREA. l6l Now, in passing to polar co-ordinates, let r = PFbe the radius vector, and AB the initial line. Then we have x — r cos v, and y — r sin v. Substituting these values in (15), and per- forming the indicated squaring, we easily obtain by transpos- ing, observing that sin^ v -\- cos^ z^ = i , . r" — 2aC — C Ar2r(a — C) cos v = 2aC— C'-^2rVd' - 2aC-\-C' cos ^. (16) Now put 2c for — 2aC-\- O, and also put for cos v the sine of its complement, v'. Then transposing the second member of (16), and putting -j for v\ or the angle DPY, it becomes r" — 2r V a" -\- 2c sm V -\-2c = o \ (17) and by assuming any other initial, as FG, it is plain that the present v will become v plus some constant, say g. ■ 127. In this problem the terms at the limits, which should be present a marked peculiarity. For, since the curve is to be closed, we must consider the limits of integration, viz., o and 27r, to be fixed, so that the terms become merely P^ Sr^ — P^ Sr^. Moreover, r^ and r^ become one and the same radius vector, and the terms at the limits therefore vanish without causing dr^, Sr^, Pj, or P^ to vanish. Hence these terms furnish no conditions for the determination of the arbitrary constants which enter the general solution. These constants, therefore, with the exception of a, which is fixed when the length of the curve is assigned, must remain undetermined. But this should not be otherwise. For we see from the last article that g is numerically equal to the angle YPF, while c depends upon the position of the pole with relation to the centre ; and we can l62 CALCULUS OF VARIATIONS. evidently, without affecting the resuh, assume any pole and any initial line we please. If, however, we had required that a curve of given length should pass through two fixed points, and should, together with the radii to these points, include a maximum area, the three constants would be determined from the assigned length of the arc, combined with the two equations which would hold in order that it might pass through the two given points. In leaving this subject, we may remark that whatever has been shown concerning the general treatment of the limiting terms in problems of rectangular co-ordinates will be equally applicable here. Thus, if the limiting values of v only be assigned, while those of r, r' , etc., are subject to variation, we must equate the coefficients of ^r^, ^r/, dr^, dr/, etc., severally to zero. If it become necessary to vary the limiting values of V also, we change v^ into v^ -\-dv^, and v^ into v^-\-dv^ ; and if the required curve is to have its extremities upon two other curves, equations (lo) of Art. 69, or the more simple equations (2) of Art. yOy will be appHcable w^hen we put v for x, r for j, r' for y , etc. Section VIII. DISCRIMINATION OF MAXIMA AND MINIMA {JACOBUS THEOREM). 128. We have already seen that, in discussing the maxi- mum or minimum state of any definite integral, we must equate the terms of the first order in its variation to zero, and then, having solved the differential equation obtained thereby, this solution must, if it do not reduce the terms of the second order to zero also, render them positive for a minimum and negative for a maximum. We have also seen that the method JACOBTS 'theorem. 163 of transforming these terms, so as to render their sign evident, has been far from uniform, while in some cases we have been unable to investigate the sign of these terms at all. We now proceed to explain Jacobi's Theorem, which gives us an invari- able method of investigating the sign of these terms when the limiting values of jt, 7, y' , etc., are fixed. But as the general discussion is somewhat abstruse, Ave shall begin with the most simple case, which is also the one which will most frequently present itself for consideration. Case i. Assume the equation u=iydx, (I) where V is any function of x, y and y' only. Then to the second order, inclusive, we have •^•^o {dy -^ ^ dy ^ ) the Hmiting values of x being fixed. Now the terms of the first order, when transformed in the usual manner, become P. ^y. - P. ^lo + r> ^7 dx, where dy' ' dx dy dx dy'' But if we would render U a maximum or minimum, the solution of our problem must be the value of y obtained by 164 * CALCULUS OF VARIATIONS. completely integrating the equation M = O] and since this is an equation of the second order, this value of y will certainly be some function of x and two arbitrary constants, say y=f{x,c,,c,) ^f. (3) Of course other constants may enter F, and consequently y, but with these we are not now concerned. Then, since the form of the function / will be independent of the conditions which are to hold at the limits, we must next so determine c^ and c^ as to satisfy these conditions, and then the solution be- comes complete so far as the terms of the first order are con- cerned. 129. The foregoing considerations will prepare us for the discussion of the terms of the second order ; but before enter- ing upon the explanation of Jacobi's Theorem, we may say that its object in the present case is to put the terms of the second order under the form -— j multiplied by the square of a cer- dy tain function, and also to determine the form of this function. Now, since the terms of the first order must vanish, there remain only terms of the second and higher orders, and we may, to the second. order, write SU^- fj\^^/ + 2bdy 6/ 4- cSy^) dx, ' (4) where a, b and c have the values shown in (2). Let as assume that (5y^, Sy^ are zero ; then we shall first show that d^^can be written thus: where A and A^ are variable functions, the suffix i having JACOBPS THEOREM. 1 65 no reference to limits. Observing that ^y = -3—, we have, ax by parts, fcd/'dx = cSy'dy - f Sy -^ cS/. dx, (8) Also /■ b^y dy'dx = bSy"^ — j dy -— bdy.dx = ^^/ - f^^^y ^y'^i'^ - f^ ^fd^' (9) Hence 2fbSy Sy'dx = bdf - f'^ S/dx, (lo) Therefore, collecting results, arranging and factoring, we have tf ;/ = 1 j (M/). - (M/). + {c6y d/\ ^ {cSy 6/1 } which, when we make Sy^ and (^J/o vanish, gives (5^^ in the re- quired form, and dx 130. We will now show, in the second place, that if we vary M, we may also write 6M=AdyJy-^Afiy\ (12) dx ^ ^ We have . dy dxdy' dx l66 CALCULUS OF VARIATIONS. Varying the first term, we have aSy -j- b^y' ; and varying P, we dP obtain d^y + cSy' , Hence the variation oi — -j- (that is, the change which it undergoes from changing y into y -\- dy^ and y' into y + dy'^ is — -7- {bSy -\- cdy'), or, by differentiation, — b^y' T- ^y r- cdy\ dx dx Collecting and arranging, we have and therefore we may, if Sy^ and Sy^ vanish, write 6U=\lyM6ydx. (14) 131. We have already shown that if the terms of the sec- ond order in df/ vanish, we shall be obliged to examine those of the third; and as these will not usually vanish, but will be positive or negative at our pleasure, we shall be, in general, safe in assuming that in this case we have neither a maximum nor a minimum state of U. But it is evident that the quantities A and A^ are not at all in our power, so that unless those quantities vanish of themselves the terms of the second order can only be made to disappear by the assumption of suitable values of Sy and dy' . Now let u be such a quantity as will satisfy the equation Au-\-—-Ay = o,- (15) dx • J A GOBI'S THEOREM. 1 6/ where u' = — -. Then it is clear that if Sj/ throughout the defi- dx nite integral can be taken equal to u, or to ku, where k is any constant, dUto the second order will vanish. Of course since dy and Sy' must be infinitesimal, k must be also infinitesimal, unless ti be already so. 132. We will next determine the quantity u, as we shall then be better able to see how it may be employed. We have seen that the value of y obtained by the complete integration of the equation M = o will be of the form y — f{x, c^, c^ = /, and that this value of y vt^ill satisfy the above differential equa- tion independently of the value of c^ and c^. If, therefore, we make any changes in the form of the values of these constants, the resulting changes in jj/ and its differential coefficients, while not necessarily zero, will not prevent these quantities from still causing M to vanish. Now suppose we change c^ into c, + dc„ and c^ into c^ -\- ^c^, where ^c, and ^c^ are infinitesimal but independent constants. Then denoting by Sy and dy the corresponding changes in y and y\ we shall have and '>''£"■+'£"■ <■«) *>' = s(f*.+|*-)' (■') Hence these values of Sy and ^y\ if admissible throughout the range of integration, will render the corresponding varia tion, (S'M, zero throughout those limits, and will also, as we see from (14), render S^U zero. But we shall find it conveni- ent to write 1 68 CALCULUS OF VARIATLONS. where k = dc^ and / =: — ^ ; and as dc^ and ^c^ are entirely in- dependent, we can make / assume any real and constant value we please. We conclude, then, from (13) and (15), that the general value of u, if not infinitesimal, is -=s;+'i- <■») But although this is the most general form of it, it is evident that we need not vary both constants in /, so that we may have ku = -^'^c, or ku=^ -^ dc^. (26) 133. Let us next consider whether ku can be an admissible value of ^y throughout U\ because if it can, there will be no need of any further transformation of the terms of the second order, since there will be at least one mode of varying j which will cause these terms to vanish. We observe, first, that since dy and Sy' must be always in- finitesimal, if ku be an admissible variation of y for any por- tion of the integral, say from x^ to x^, u and u' must remain finite throughout these limits. In the second place, if ku be an admissible variation of y throughout a portion only of the required curve, say from x^ to x^y while the values of x^,y^, x^,y, are fixed, then to certainly make the terms of the second order vanish we must have y^ and JK3 also fixed ; must change y into y -\- ku throughout the limits x^ and x^, and leave the rest of the required curve un- varied. As this requires that u shall vanish, both when X = x^ and when x = x^, and as dy could not equal ku through- out any limits unless u vanish at both those limits, we con- clude generally that to make the terms of the second order J A GOBI'S THEOREM. 1 69 disappear by the use of kii for 6y, u must vanish at least twice within the limits of integration. In the third place, if either of the quantities -4- or -^, which are not in our power, vanish twice within the range of inte- gration, while at the same time its first differential coefficient with respect to x remains always finite, we can make the terms of the second order disappear by putting that quantity for ?/, but not otherwise. Moreover, that we may employ the general value of kti^ all the quantities ^, - — ^, -^ and — — ^ must remain finite dc^ dx dc^ dc^ dx dc^ throughout the limits for which ku is employed, and we must also be able to so assume ti that it shall vanish at least twice as we pass from x^ to x^. We will now consider under what circumstances this lat- di ter condition can be fulfilled. Put h for — ». Then we see df_ dc^ from (19) that we can cause u to vanish for any value of x we please, say for x = x^, by taking / = — K/, and this is all that we can effect. We can, moreover, in some cases assume u so that it shall not vanish as we pass from x^ to ;i-,, while in other cases we cannot. For our power over u depends en- tirely upon our assumption of /. Now suppose we find that Ji, which is not in our power, cannot assume all possible values from negative to positive infinity as we pass from x^ to x^. Then, by assuming / equal to one of these values, but multi- plied by — I, we can effect that u shall not vanish within the Hmits x^ and x^. But if, on the other hand, we find that h ranges through all real values, we cannot assume / so that u shall not vanish at least once. To apply the foregoing, assume / so that u shall vanish when X — x^. Then if the range of h through all real values I/O CALCULUS OF VARLATIONS. be complete, il will evidently vanish again at or before the upper limit, according as h may complete or more than com- plete its range, and we can make the terms of the second order vanish by the use of kit. But if the range of Ji be only partial, u will not vanish again at or before the upper limit, and we cannot employ ku to make those terms disappear. (34, It is evident that when kit cannot be employed to make the terms of the second order vanish, some further trans- formation will be necessary to render their sign apparent ; and to this we now proceed. Let u involve k — that is, be ku — so that it may be infinitesi- mal, and resume the equations ^u^\Jiy\^^y^^^^fiy'\^y'^^ (21) and Au-{-4-A,u' =^0, (22) ax Then whatever be the value of (^^', we may certainly make it equal to ut, and (21) will then become where hii)' = -— ut. ax We wish now to reduce (23) by integrating it by parts ; but before doing so we must show that because (22) is true, the expression u - AiitA^-—A,{ut)' I dx or Wdx (24) can always be integrated, its integral taking the form B^t' , where B^ is a new variable function, the suffix i having no ref- reference to limits, and t' = -—. dx J A GOBI'S THEOREM. I7I 135. Multiply (22) by tit, and subtract the product from the value of W in (24), and we have W^u\±^Aiut)'-ut£-A,.'\. (.5) Now u — A^itit)' — —- uAltif)' — Alutyu. (26) But {lit)' = uf + ///. Whence u4-Alut)'=. ^u'A.t'^^ —tiA.tu' -A.u'ut' -A.u'H (27) ax ax ax and Whence ——itA^u't = iiA^t' -\- 1 -— uA^u' . ax ax u A. ASuty = I -^ uAy - Ay \t+-f: ^''^^^'- (2^) ax \ ax J a.X' Now if the differentiation indicated in the first member of (28) were performed, it is evident that the only term in which t could appear undifferentiated would be ut-^Ay or \4-uAAi' -A.iiAt, ax { dx ) Hence we see from (25) that the terms in li^ which contain t will cancel, and we shall have V'^^ic'A,t' = ^Bjf, ax dx where B,= n'A, (29) and / Wdx =/^_ B,t'. d^ = B/, (30) the constant being neglected. 172 CALCULUS OF VARIATIONS. 136. By the use of (30), (23) may now be integrated by parts thus : Wtdx 2e/Xo (31) = \ \ ^tB/\ - {tB,t\ I - \iyrdx. Now examining equations (29), (11), (4) and (2), we see that d'^V B, = u'A, = -u'c=-.^u''; (32) and since we put Sf equal to ut, we have ,^usy-^^ (33) If the terms without the integral sign in (31) do not vanish, they must be added to those already in (11). But the suppo- sition that 6j/^ and fy^ are zero will certainly reduce these terms to zero unless 21^ and u^ vanish, which would, as we have seen, indicate generally that there is neither a maximum nor a minimum. Therefore, finally substituting for B^ and /' their values from (32) and (33), we have d/' u' dx -2eAo d/^ u^ ^-^' (34) and if we now consider ii as no longer involving k, we must multiply the last member by k^, 137. Let us now consider the last equation more particu- larly JACOBPS THEOREM. 173 First. We shall assume that before obtaining this equation it had been ascertained that the terms of the second order could not be reduced to zero by any use of ku for Sy ; that is, that u could be so assumed as not to vanish at all, since other- wise the last transformation would be needless. / Second, Now suppose the second factor of (24) does not vanish permanently, in which case it will evidently be posi- tive ; and also that it remains finite throughout the range of integration. Then for a maximum or a minimum we require d'^V only that -— ^ or c shall remain finite, shall not vanish perma- nently, and shall be of invariable sign. For we have already seen that infinite values cause the method of development em- ployed to become inapplicable, and even in the case of a single element of an integral, render the entire result doubtful. More- over, if c can change its sign, we can, as has been previously shown, vary y for such values of x as will render c negative, while leaving y unvaried for all other values of x, and thus make d^ negative ; or by pursuing a similar course with such values of x as render c positive, we can make dU positive. But if c remain finite, be of invariable sign, and do not vanish permanently, we shall have a maximum or a minimum accord- ing as it is negative or positive. Third. But suppose the second factor of (34) does vanish. Then we must have Whence u'Sy — udy' — O. (35) u' J Sy' . dti dSy — ^;r = -f- dx, or — = — -^ . u oy li dy Therefore ISy = lu -\- g— lu-^ Ik— likii), and dy — ku, where k is any infinitesimal constant. But by supposition the prob- lem is such that Sy cannot be made equal to ku throughout the range of integration, and therefore the second factor of (34) will not vanish permanently. 174 CALCULUS OF VARLATLONS. Hence we see that if the terms of the second order cannot be reduced to zero by the use of ku, then unless c vanish they cannot be reduced to zero by any admissible mode of varying y, and this supplies what was before wanting in the complete investigation of the subject. To render the second factor of (34) infinite, we must, if dy and Sy' be infinitesimal, have either zi =0 or u' — 00 . But the first condition disappears, since we suppose zi to be taken so as not to vanish at all, and the second cannot occur unless ^- ^ or -— ^ become infinite. ax dc^ ax dc^ It will be seen that the expression uSy — Syti' m (34) is the determinant of u, u' , Sy^ dy'\ so that, putting D for their determinant, we may write 2 <^^o u"" ' and we shall see hereafter that determinants can always be employed in expressing the final results of Jacobi's transfor- mation. 138. Before applying this theorem to any example the following general directions may be useful. First. Having obtained the general solution, find — ;- or c, dy which must not vanish permanently, become infinite, nor change its sign. For in the first case the terms of the second order would reduce to zero; in the second the investigation would become more or less unsatisfactory ; while in the third the terms of the second order can be made to assume either sign, thus rendering a maximum or a minimum impossible. Second. If these conditions be satisfactory, find the quanti- ties -— and -5^, neither of which must vanish twice within the dc, dc„ JACOBPS THEOREM. 175 range of integration, otherwise we can reduce the terms of the second order to zero by employing this quantity for u. Third. Moreover, the first differential coefficients of these quantities with respect to x should remain finite as we pass from x^ to x^, otherwise some element of SU mscy become infi- nite, thus rendering the result untrustworthy. Fourth. If all these conditions still indicate a maximum or a minimum, consider next whether, in the general value of //, h or the ratio between the quantities -4- and -4- can rang-e over dc^ dc„_ all real values as we pass from x^ to x^. For if it can, the terms of the second order can be made to vanish by the use of ku ; but if it cannot, those terms cannot be reduced to zero by any admissible values of dy, and our investigations are complete, assuring us of a maximum or a minimum according as c is negative or positive. Problem XXIV. 139. It is required to apply Jacobi s Theorem to Prob. I. Here the general solution is y = f{x, c„ e,) = f= c,x + ^,. (i) Also, so that d'V I dy |/(i_|.yy (2) and this last expression is evidently positive, finite, and of in- variable sign. We likewise obtain from (i) 176 CALCULUS OF VARIATIONS. df and ir = i» (4) and df _ dc. I, d df _ dx dc^ I, d df _ dx dc„ 0. (5) (6) Now neither of the first two quantities can vanish twice, nor do their first differential coefficients become infinite. More- over, if we divide the first of these quantities by the second, we find h = x, which will not range through all real values. Hence u can be so assumed as not to vanish at all. For we have zi ^^ X -\- 1] and bv assuming / to be negative and numeri- cally greater than x^, the truth of the assertion becomes evi- dent. Jacobi's Theorem, therefore, indicates a minimum in this case. Problem XXV. 140, // is required to apply the theorem of J ac obi to the case of the brachistochrone in Prob. II., Case i. Here, from equation (11), Art. 17, the general solution, which is a cycloid, is seen to be of the form y —f{x, c„ c^=f— c, versin- - -- V2c^x — x'' + c^, (i) where c, is the radius of the generating circle. We also have Vx so that ^^= _ ' (2) J A GOBI'S THEOREM. 1/7 This last expression is of invariable sign and positive, but be- comes infinite at the cusp, where both x and y' are zero. The investigation will therefore be subject to any doubt which may arise from this fact, (See closing remark of Art. 21.) Disregarding this objection, we have from (i), by differentia ating carefully with respect to c^ and c^ successively, while treating ^ as a constant, -f- = versin , (3) df_^ dc„ (4) '2 Now we shall take x^ to be somewhat less than 2c^. For, as we have seen, y' becomes infinite at the vertex, and we wish as far as possible to avoid infinite quantities, since Jacobi's method does not enable us to overcome the obstacle which these quantities present to a satisfactory solution. With this limitation neither of the above quantities will vanish twice within the range of integration. We also have, by differen- tiating in the usual way, d df _ -\ri ^^, dx dc^ {2c ^ — x)^ d df _ dx dc„ o, (6) and these quantities remain finite throughout the present lim- its. Moreover, if we divide -f- by -/-, the quotient /i will be dc^ dc^ the second member of (3), and this cannot range over all real values, so that ii can be so taken as not to vanish at all is we pass from x^ to x^. We conclude, therefore, that, setting aside the objection previously mentioned, Jacobi's Theorem indicates a minimum in the present case. 1/8 CALCULUS OF VARLATIONS. Problem XX VI. 141. It is required to apply the theorem ofjacobi to Prob. XXI I. From what has been previously said regarding the treat- ment of polar co-ordinates by the calculus of variations, it will appear that all the reasoning by which Jacobi's transformations were effected will apply also to them when we change x into v^ y into r, and y' into r' . We shall consider only the case in which we have an ellipse, our object being to verify the clos- ing remark of Art. 122. We shall, with slight deviations, fol- low Prof. Todhunter. (See his Researches ; or Adams Essay, Art. 183.) Here, as we see from equation (5), Art. 121, V=V--- Vr' + Whence i/l-L dW ^ r a dr" ^{r'^r'y which cannot change its sign, and is always finite and positive. Now the general solution in equation (15), Art. 121, may be written v/ X /- ail — e^) , . .=/(.,..,.,)=/=-^^-^, (I) where e may take the place of c^^ and g that of c^. It appears that (i) contains also another constant, a. But this constant was introduced when we assigned the initial velocity, and is not therefore a constant of integration. Now we have already stated that /might involve, besides the inde- pendent variable and c^ and c,^, any number of other constants; JACOBVS THEOREM. 179 those only which enter by integration being considered by Jacobi's method. We must, then, pursue the usual course, and find the dif- ferential coefficient of /, that is, of r with respect to e and g. We have, from (i), ^SLufl = I 4- ^ cos {v + g), (2) r Now differentiating with respect to e, we obtain r r" de _ ^ ' ^ ^ e { r ) the last member being found from (2). Solving (3), we finally obtain and Also, dr _r^ — ar{i -\- e^) de ae{i — e") d dr _ [2r-a(i+e')y dv de ae{\ — '/) (4) (5) dr _dr _ , dv dg ^^^ Now neither the first member of (5) nor (7) can become infi- nite, so that we may employ Jacobi's Theorem with confidence. But before resorting to the most general method, let us determine whether the first member of (4) or (6) can vanish dr twice. Now to make — - vanish, we must have de r = «(i + .'). (8) l80 CALCULUS OF VARIATIONS. But this is the value of the radius vector drawn to the ex- treraity of the remote latus rectum. For the distance between the foci being 2ae, and the semi-latus rectum being a{\ — e^), we have ^^ ^ 4^V" + ^^ (i - ej ^^a'^i^ ej. dr Also r' , and consequently -— , vanishes at each vertex of the dg ellipse, so that we conclude at once that there will be no mini- mum if the arc extend from vertex to vertex, or be cut off by the remote latus rectum. Now, in applying the general method, we are only con- cerned in knowing the range of h, or the ratio of ; to -^. dc^ dc„ But h evidently varies as But Whence • 3—^- or , (9 r r I __ I 4" ^ cos {v + g) r a{i-e^) r' esm(v+g-) r' a{i-e^) ' and therefore the last member of (9) may be written i-^(i+.o ^sin(^4-<^) (10) J A GOBI'S THEOREM. l8: Now this expression varies only as (II) sin {v-^g) rsm{v-\-,g) Next let us write r = 2a — R, (12) rsin(^ + ^) = i^sin w. (13) Then R will be the radius vector drawn from the other focus, and w will become the angle which R makes with the major axis. Then, by substitution, (11) will become p . = -. ] -^-^ — - - il^ecotw, (14) Rsmw sm ^ ( K ) the last member being obtained by substituting for R its value — ^^ —, whence h varies as cot w. I -\- e cos w Now, in general, any function will have a complete range from negative to positive infinity when we can cause it to start with a given value, change sign by passing through zero or infinity, and return to its initial value. But cot w be- comes infinite at the two vertices only, vanishes only when r is the semi-latus rectum, and changes sign at these four points, and at these only. Now let i?o and R^ be the radii drawn to the two fixed points. Then, to make cot w^ and cot w^ equal, r^ and r, must form a continuous line ; that is, a focal chord. Should the arc extend from one vertex to the other, cot w^ and cot w, will not be equal, but will be infinite and of contrary sign, having passed through zero. But in all other cases cot w^ and cot zv^ are equal, after having changed sign by passing through in- finity. ^ l82 CALCULUS OF VARIATIONS. Here, therefore, there is no minimum, and if the arc be still greater the same remark will hold, unless we were required to vary the entire arc. For since we can make it vanish at each end of the focal chord, we can take dy = ku through that portion of the arc, and leave the remainder unvaried, thus making the terms of the second order in (5^ t^ vanish. But if the arc be less than that subtended by a focal chord passing through the present, which is the remote focus — that is, both foci lie upon the same side of the line joining the two fixed points — then the range of cot w will be only partial, and there will be a minimum. 142. We may give a general geometrical illustration of Jacobi's method. Let A and B be two fixed points, joined by a curve which satisfies the differential equation J/ = o, and let CED be another curve derived from the first by such vari- ations of y and y' as will result from varying the constants of integration, and consequently still satisfying the same differ- ential equation. Then there will, if -r-w permit, be a maximum or a mini- dy mum when CED cannot twice meet AB unproduced. But if it can meet it twice, we may regard AFEGB as the new de- rived curve, which would make the terms of the second order vanish. But since we can make u vanish once at pleasure, we may suppose the derived curve to touch the other at A — that is, we can make C and A coincide — and then we shall have a J A GOBI'S THEOREM. 1 83 maximum or a minimum so long as the other point of meet- ing, G, is not reached. Moreover, we compare AB with such derived curves onl}^ as satisfy the equation M — o, ahhough their number may be infinite. For we have seen that when ku cannot be used to make the terms of the ^second order disappear, they will not 72 T" vanish at all if -—^r do not vanish. Hence no other class of ay curves could render SU \.o the second order zero. (4-3. Now it is evident that, in order to employ the pre- ceding theorem, we must be able to find the functions -— and — ; that is, to determine the change which y would undergo when in the general solution we give infinitesimal increments to c^ and c^. We therefore naturally first seek to obtain the complete integral of the differential equation J/ = o, and to exhibit it under the form of j/ =zf{x^ c^, c^. But it frequently happens that even when we are unable to obtain the general solution in the explicit form just given, we can still determine the functions -^ and -^. Still this is not dc^ dc^ strange, since we can often obtain the differential of an un- known quantity ; that is, a differential whose integral is unob- tainable. When these functions can be found, Jacobi's method can be applied to the inves,tigation of the terms of the sec- ond order, whether the equation M = o can be completely integrated or not ; and we now proceed to show how they may be determined in the case of a very important class of problems. The following method is due to Prof. Todhunter (see his Researches, Arts. 26, 282), and we shall see that by it he has been able to obtain some results not previously known, and to correct some which had been erroneously given. 1 84 CALCULUS OF VARIATIONS. Problem XXVII. 144-. It is required to discuss in full the conditions which will maximize or minimize the expression U= r^y^ V dx = r^ Vdx, Xa where v is any function of y' and constants. Here F is a function of y and y' only, and dy Hence, by formula (C), Art. 56, we have whence f'{v-y'v')^c,, (I) which is as far as the integration can be carried, so long as n and V are entirely undetermined. But we may suppose a curve to be drawn satisfying (i), and that its equation is y z=i fix, c^, c^ =/. Then, although we cannot determine the form of/, we can ascertain what would be the corresponding variation oi y if c^ and c^ were increased by Sc^ and Sc^, and can then investigate the terms of the second order. 145. From (i) we have Also, ;'=f-^=/(/>0=/. (2) v — y v y dy' JACOBVS THEOREM. 1 85 Whence, by supposing the integration performed, we may write X = F{y\c,) +c, = F-\- c,. (3) Now, although / and F mayjcontain other constants besides c^, these will not be affected by any variation of c^ or c^, leaving only y' and c^ as variables. Moreover, x will undergo no change when c^ and c^ vary, and these constants themselves are entirely independent of each other. We have then, from (2) and (3), dc, dc,~^dydc, ^^^ and Whence _dF dFdy ^~ dc,'^~dy'~d^^' ^^^ d_ld_l ^_dF^ dy' dc^ dc^ ^ ' Differentiating (2) and (3) with respect to x, we obtain and Whence Hence, and then multiplying by (8) and comparing with (6), df dy' _ y' dy' _ y'dF dy' _ y'dF dy dc^ /' dc, dy' dc, dc. (10) 1 86 CALCULUS OF VARIATIONS. Therefore dy _df y'dF dc^ dc^ dc^ ' Again, from (2) and (3), we have dy _df dy' dc^ dy' dc^ and 00 (12) dFdy' , , , dy Whence, by (9), Comparing this equation with (8), we obtain -^ — — y' dc„ !;=-/• ('4) We must next determine the form of f~-J and f-^J, which are only partial differentials with respect to c^^ this fact being indicated by writing them in brackets. From (2) we have where m — -. Hence n \dcj {y — yv'f ^ ^ But from (i) we have ^ m r m V -^ J ymn y ' and therefore, restoring n, w^e have dcj ncl (17) JACOB/' S THEOREM. 1 87 Now although we cannot, while v is unknown, determine F, still it is evident, from its mode of derivation from /, that if Cj^ enter the latter as a factor, it must also enter the for- mer unchanged. F must therefore be of the form c;"^w, where w is some function not involving c^ or c^, but merely y\ and perhaps constants, not of integration. Hence, from (3), we have ' x = c,^w-{-c^ (18) and X — c^ w = -. Now Therefore, finally, dF^ = wmcJ^-^ — -, (10) dy y — y'{x — c^ dc, nc^ (20) (46. Now if the value of y found by the solution of (i) can render U a maximum or a minimum, the terms of the second order in dC/can be put under the form given in equa- d'^V y'^d'^v tion (34), Art. 136. Then, supposing -— ^ or —r-jr ^^ ^"^'' ^^ be of invariable sign and finite, it will only be necessary that u shall be incapable of vanishing twice ; which will in general, as we have seen, follow if it can be so taken as not to vanish at all. Now equations (14) and (20) give us the general value of 2/, thus : dc^ dc^ dc^ dc^ dc^ where L = ncd. 1 88 CALCULUS OF VARLATLONS. Now by differentiating the last equation with respect to x, it will at once appear that u' will not become infinite so long as y" is finite — that is, so long as there occur no cusps. Were this not so, we could not feel entire confidence in the follow- ing investigations. But in order to make u vanish without supposing either of the quantities -^ or -f- to vanish, we must have ;tr — ^ = r, — Z. (22) Now if y be the ordinate of the curve, we know that the first member of (22) will represent the abscissa of the point in which the tangent to the curve at y will meet the axis of x, and we will denote this abscissa by X. But since Z is a constant entirely in our power, we can give to the second member of (22) any value we please. If, therefore, there be any real value which X cannot assume, we can, by making the second member take that value, render equation (22) impossible, and thus cause that u shall not vanish at all. But suppose either of the quantities ~- or. -5- to vanish twice. Then equating the first to zero, we obtain x — -^ = c^. Whence, if it vanish twice, there must be two tangents which meet on the axis of x at the point whose abscissa is c^. That the second quantity may vanish twice, y must also vanish twice. (47. We may now complete the discussion of Prob. VIII., as promised in the closing remark of Art. 63, Here n is unity, and /of that article is identical with t/. Suppose, as before, that y is positive, but that the curve, in- stead of being concave, is always convex to the axis of x. Then X cannot always range over all real values. For sup- J A GOBI'S THEOREM, ■ 1 89 pose the line AE to slide as a tangent along the curve from A to B. Then if we assume DE as the axis of ;r, this line cannot meet x between D and E, and the range of X is not therefore complete. But if CE be the axis of ;r, X will assume all real values, its range being just complete ; while if GH be taken as the axis of ;r, then X, having passed through infinity, will complete its range before B is reached, and will then repeat the values of ;r from G to N. If we consider such an arc as BK, the range of X will evidently be restricted, and the tan- gents at B and K will intersect above K — that is, above x — since the ordinate of K must be positive. ^^~\ Hence when y'^ is positive, if the tangents at the extremi- ties of the arc intersect above the axis of x, we shall have a maximum or a minimum according as v'^ is negative or posi- tive, because j/ is positive, and we have seen (Art. 63) that wheny^ is of invariable sign, /^', which is here u'^ will be also. But if the extreme tangents intersect on or below the axis of X, there can be neither a maximum nor a minimum. Problem XXVIII. 14-8. B is required by means of the preceding method to apply Jacobis Theorem to Prob. VI I. Here the general equation to be considered is U= r^yVT+Y'dx^ r^yvdx. 190 CALCULUS OF VARIATLONS. Whence v" = -_===^^y a positive quantity ; and as the gen- eral solution is a catenary, having the directrix as the axis of X, y is always positive. Therefore we infer that the solution will render U a minimum Avhen the extreme tangents intersect above the axis of x, but not otherwise. Suppose, then, the same condittons and notation as in Art. 61, which will of course hold even should j^ and j^ become equal. Now the equations of the extreme tangents are J/ — I; = J// (jr — c) and J — /^ = jFo' (-^ + ^). From these equations we obtain W y — k — y,'c' and solving for y, and giving it a suffix, because it will then be the ordinate of the point of intersection only, we have '- — jr^' — • ^"^ Now put 2c _2c L — e'^ — e «. (2) I - Then multiply equation (4), Art. 61, by y ^", equation (5) by I _ ^ je~a^ subtracting the second product from the first, and then, observing that the first member of the resulting equation be- comes identical with the second member of equation (i) of the same article, we have, as the equation of the catenary, y= }c^lb(^ — ke ~''\+c '"'Ike'' — be « j ^ (3) JACOBPS THEOREM. I9I Now differentiating (3) with respect to x only, and then sub- c ex stituting successively in the result e^ and e ^ for ^, we have . Mb-2k (4) (5) where Therefore But yr = La ' v' — 2b -Mk Jo — La ' M=^ 2c 2c (6) f/-W={M-2){d + k)±. (;) Z^ = J/^ - 4 = {M+ 2) {M- 2). (8) Whence M— 2 must be positive ; and as L cannot become negative, (7) must also be positive. Multiplying (4) by k, (5) by b, and subtracting, and then multiplying (4) by (5), we have the equations ky, - by^ = ^; ^ (9) and ^ 2M{b^-\-k^)-bk{^ + M^) _ 2M{U^ + k^) - bk{A + 2M^ - M^) ~ zv ^ ^ Multiplying (10) by 2^, adding to (9), reducing to a common denominator, and factoring, we have 2cy:y,' + ky,' - by,' = ^\{b'-\-^ - Mbk) (2Mc -La) + {M' - ^) cbkX . (11) 192 CALCULUS OF VARIATLONS. But performing the multiplication indicated in the second member of equation (6), Art. 61, it may be written ^^Mbk-{b'-^k'). (12) Hence, and recollecting that M'^ — 4= L\ the second member of (11) will become La ,, . 2cdk . ^ -^Mc + — . (13) But equation (8), Art. 61, may be written and hence, since L is always positive, the sign of (13), and con- sequently that of j/2, the ordinate of the point in which the extreme tangents intersect, will be like that olF\ Now it was shown that when but one catenary can be drawn, F^ is zero ; but that when two catenaries can be drawn, F^ will be positive for the upper and negative for the lower. Hence the extreme tangents to the upper catenary will inter- sect above the axis of x, thus giving us a minimum ; while those to the lower will intersect below that axis, and will not give a minimum. When but one catenary can be drawn, the extreme .tangents will intersect on the directrix, and we shall not have a minimum. Indeed, we may here suppose that the two catenaries coincide ; and for a demonstration of the fact that the extreme tangents w^ould in this case intersect on the directrix, see Todhunter's Researches, Art. 72. JACOBVS THEOREM. 1 93 Problem XXIX. 149. // is required to apply the general method of Art. 146 to Case 2, Prob. II. Here n= and v= Vi +J^ so that 2 ^"V „ „ I dy |/_^(i j^yy which is always positive and finite ; thus indicating a mini- mum, so far as it is concerned. Now as the general solution in this case is a cycloid, having the horizontal as the axis of x, we know that X cannot assume all possible values, since no tangent can meet the axis of x within the cycloid. Hence, without determining jj/ as a function of x, or even obtaining the value of u, we are able easily to apply the method of Jacobi, and to see that we have a minimum. This result is, however, subject to any doubt which may arise from the fact that y is infinite at either cusp, but is alto- gether trustworthy so long as the portion of the curve which we are considering does not contain any cusp, as will be the case if the particle is to start with an initial velocity. Problem XXX. (50. It is required to apply the theorem of Jacobi to Prob. XVL Here, as will be seen from equation (8), Art. 98, the gen- eral solution is a sphere, having its centre upon the axis of x ; and, recollecting that y must not become negative, that equa- tion may be written y=VAd'-{x-c:f. (I) 194 CALCULUS OF VARIATIONS. Now it must be observed that a is not a constant of integra- tion, but was introduced in accordance with Euler's method for treating problems of relative maxima and minima, so that it cannot be varied in applying Jacobi's Theorem ; and func- tions involving it, together with x, j/, c^ and c^, will merely be mentioned as functions of the latter quantities. It appears, then, that y has in this case been obtained mere- ly as a function of x and c^, it having been necessary in equa- tion (3), Art. 98, to make the first constant of integration zero before we could effect the second integration. Since, there- fore, the constant c^ has disappeared from the value of j/, we dy dy shall not be able readily to obtain the functions -f- and -7- re- dc^ dc^ quired in the application of Jacobi's Theorem. 151. Since we have seen (Art. 99) that the sign of 2a must be negative, we have from equation (i), Art. 98, F = /- 2ay^/\^y'\ Therefore (rV_ 2ay dy^~ V(i+/7' which, being negative, indicates, so far as it is concerned, that the volume is a maximum. Now observing the sign of 2a, equation (3), Art. 98, may be v/ritten -^^==/ + ^. (2) Vi-\-y'' But from (2) we see that / can be expressed as an explicit function oi y and c^\ and we have always dx or -^=f{y,c,)dy, (3) y JACOBVS THEOREM. 195 Whence, supposing the integration to have been performed, we have '*^=/UO + ^.=/+^.- (4) Therefore — must in any case equal — . Taking the total dif- dy y ferential of (4) with respect to c„ recollecting that any change in c^ will affect y but not x, we have Hence dc^ dy dc^ dc^ y' dc^ ^ = -y^. (6) dc^ dc^ Now in hke manner, recollecting that c^ does not occur ex- plicitly in /, we have and therefore dy dc^ y dc^ %.'-"■ <'■' We must now determine the value of -—-, observing that it is dc^ only the partial differential coefficient of /with respect to c,. If / could be found as an explicit function of y and c^, this could be done directly ; but as / cannot be so found, we must adopt an indirect method. Now the supposition \\\'dXy is to become constant, and c^ variable, will make dy constant, but y' still variable, because it is capable of being expressed as an explicit function of y and <:,, although -^ will be no longer total, but merely partial, and can be at once found. But 196 CALCULUS OF VARIATIONS. f— /— 7-; and if in this expression we vary ^,, regarding jj/ as y constant, and indicate partial differentials by brackets, we shall have But in this case we must have dy=z -^ dc^\ and as 6c^ must be constant, we have rdf- Now from (2), by partial differentiation, we obtain 2ayy' [dy'^_ 4/(1 +y^) Hence 152. When the general solution is a sphere, this integral can be obtained. For if in (2) we put r for 2a, make c^ zero, and divide by y, it will become the differential equation of the circle, whose centre is on the axis of x ; and we shall have and dy^ '^''^^ Hence (10) becomes [l]=-r/^^>- JACOBVS THEOREM. 197 dzv Now put y = tan w and dy' = r — . Then cos w rdfl _ I f dzv _ ^ r cos' w + sin'' w , \_dc^j r^ cos w sin' w r^ cos w sin" 2x/ I j f* cos w , ' r dw \ . = 1 / -^-^dzv-\- \. (12) r ( *^ sin zi; ^ cos w ) ^ Now by integrating this expression, we shall obtain r^l = -ii I I ,i +sin7c> ] ^ _i_^^ U/^ J ^ I sin 2£/ ' 2 I — sin zt/ j r * Hence, finally, by equation (6), we have ^ = --?^' Z. dc^ r (14) It will then at once appear, by comparing (7) and (14), that the range which we are in this case to examine will be entirely dependent upon that of Z. Now when zt/ is -, Zis — 00 ; and when w is zero, ^ is + °^ *» so that Z ranges twice from — 00 to -|- 00 as we pass from x^ to x^. We would therefore naturally infer, from the employment of Jacobi's method, that the sphere is not the solid of revolution whose volume for a given sur- face is a maximum ; an inference which we know to be erro- neous. 153. Although for convenience we have hitherto tacitly assumed that, even when the terms of the second order are to be considered, we may by Euler's method convert any prob- lem of relative maxima or minima into one of absolute max- ima or minima, we have not yet established the correctness of this assumption ; while we see from the last article that it can- 19^ CALCULUS OF VARIATIONS. not be universally true. In order to discuss the subject in a general manner, let us resume the conditions and notation at the beginning of Art. 92. Then, as there, we shall have / 6vdx=: Vdydx and / dv' dx = / Vdydx. Moreover, since the limiting values of Sj/, dy, etc., are to van- ish, the terms of the second order will become -T' 6 Vdydx and -T'd Vdydx, This we have already seen to be the case when the func- tion contains no differential coefficient higher than y' , and we shall subsequently see that it is true generally. It must likewise be observed that now, besides being infini- tesimal, the variations of j, y' , etc., are restricted to such sys- tems of values as will render / v'dx constant ; and although U Xq we cannot express explicitly the nature of this restriction, and although the systems of values which it permits for Sy, 6y\ etc., may still be infinite in number, it cannot be disregarded in the discussion of the problem. We shall denote this restriction by writing the variations affected in brackets ; then, to the second order, we have Xyv^ dx ^/;' v\sy^ dx + \iy v] m dx=k^i (o and £ySv'^ dx =£' V'idy] dx + \S^ySV'^ \Py\ dx^m+ n. (2) Now since / 'vdx is to be a relative maximum or minimum, k -\- 1 must certainly be a small negative or positive quantity JACOBPS THEOREM. I99 of the second order ; and since / v'dx is to undergo no change when y, y\ etc., are varied, m -\- 71 must vanish, at least so far as any quantity of the second order is concerned. 154-. Thus far there can be no doubt; but what follows may perhaps be subject to some criticism, as the author has not seen it in any other v/ork, although he will not assert that no similar discussion occurs. Now the equation m = — n must be true to the second order, so that it appears that in need not vanish absolutely, but must become less than any quantity of the first order ; and we are therefore led to infer that k also will not vanish, but become a quantity of the second order. That this supposition is not inadmissible in problems of relative maxima and min- ima, we have already seen in the beginning of Art. 94. But these suppositions regarding k and m will not invahdate the V reasoning of Art. 92, by which it was shown that /or-- must be a constant ; because / could not differ from a constant by any finite quantity. Now assume the equation C"dx+b£yjdx^u, (3) where b is any constant whatever. Then, since J v' dx is to undergo no change when we vary jj/,y, etc., the variation of u to any order, as the second, will to that order equal merely the variation of its first term. Hence we may write £\sv-\dx=£;[v-^bv'\\sy\dx + \.C { \.^n+b[SV'^ \ [Sy-\dx = k^-^bm + l-\-bn. (4) 200 CALCULUS OF VARIATIONS. Now so long as b remains undetermined, k -f- bm may be a quantity of the second order ; but when, as explained in Art. 92, we put b — a — — y-, we effect that k -(- am shall certainly vanish, since those terms are then equivalent to Therefore we have ^ + '^^ = \S.l^ [\pn+'^\p n ] i^j'i dx, (5) as the exact expression to the second order of the change which / vdx will experience when y, y', etc., are varied ac- cording to the conditions of the problem ; and this is the only ' mode of rendering the expression exact, since it is not only sufficient, but also necessary, that b should become a in order to make the terms of the first order entirely vanish. Now according to Euler's method, let U be what u be- comes when b — a. Then to the second order we have \s£yod^ = \SU^ = ij("" i [cJF] +« [tfF'] I \_Sy-\ dx. (6) Whence it appears that we can and must employ Euler's method to obtain the terms of the second order in an explicit form. But it will be observed that the restriction still adheres to the variations in (6), and no method of further determining its effect upon the general form of (^C/has yet been devised; still, if, as is usually the case, the general solution can render the second member of (6) invariably negative or positive for unrestricted values of Sy, Sy\ etc., this restriction can, of course, exercise no influence upon the problem, and we shall be certain of a maximum or a minimum. But if, on the other JA GOBI'S THEOREM. 201 hand, by employing the most general values of Sy, S/, etc., it should be found possible to cause the second member of (6) to assume either sign or to vanish, we may conclude justly that t/is not an absolute maximum or minimum. But this will not warrant us in asserting that [/, and consequently t/a;o vdx, may not be a relative maximum or minimum ; that is, a maximum or minimum for all such values of Sj/, 6j/\ etc., as will render / ' v'dx constant ; and having no means of taking propor account of this restriction upon the variations, we may, at least theoretically, be unable to determine whether U is or is not a relative maximum or minimum. 155. Thus we see, first, that Euler s method must be em- ployed in developing the terms of the second order in this •class of problems ; and if by it we seem to have a maximum or a minimum, we may accept the decision as final. But if, on the contrary, we appear to have neither a maximum nor a minimum, we cannot always conclude that such is really the case, the discrimination being correct as regards an absolute, but perhaps not as regards a relative maximum or minimum state of ^. This latter result is mentioned by Prof. Todhunter (see his Researches, Art. 283) ; and evident as it is, when the former is admitted, it appears not to have been noticed by any pre- vious writer. The former result, however, is assumed by him without proof. Prof. Jellett has given no discussion of the ierms of the second order in questions of this character. 156. We can now understand why the theorem of Jacobi is not as satisfactory for problems of relative as for those of absolute maxima and minima. For example, in the preceding problem the condition that the surface is to remain constant will prevent us from making Sy invariably positive or nega- tive ; and as it must change sign, it will certainly vanish at 202 CALCULUS OF VARIATIONS. least once between the limits x^ and x^, say at the point whose co-ordinates are x^ and y^. But even if we can so select x^ that u can vanish both when x = x^ and x = x^, as we certainly can by considering a hemisphere, it does not follow that we can ma*ke the terms of the second order throughout the integral vanish by the use of kit. For when we assume dy = ku throughout the first hemisphere, we may be obliged to make some change in the form of the other also ; that is, ku may not be an admissible value of ^y unless the first hemisphere be permitted to increase or diminish its surface. Nevertheless, when Jacobi's method seems to indicate a maximum or a minimum, that indication may be regarded as trustworthy. 157. We may, in passing, notice two particular and ex- ceptional cases which may arise in the general application of this theorem. These cases appear to have befn first noticed by Spitzer. (See Todhunter's History of Variations, Arts. 173, 174.) Suppose, first, that -—^ = throughout the inte- dy gral. Now if V involve y' at all, it can, to render this equa- tion true, contain only its first power. Therefore the general form of V must be V=f{x,y)^y'F{x,y)=f^-y'F. (i) We shall write total differentials in brackets. Then U^ r' Vdx, the limiting values of x and y being fixed ; and therefore to the first order we have JA GOBI'S THEOREM. 203 Therefore, as usual, we obtain But Vdx\~ dx~^^ dy' dy so that (3) becomes df dF i-^=^- (4) Now (4) involving only x,y, and possibly constants, which are not of integration, we can, by solving for j/, obtain it as a func- tion of X without constants of integration. Hence, in appli- cations to geometry, it will be impossible to satisfy the gen- eral solution unless the given points happen to be situated upon the curve which is determined by (4). The second case is that in which we have d'V . d'V o, and dy" ' d/ Ldxdydy'} d dW-] _ o. As this case is more difficult than the former, and is rather curious than important, we shall merely give its interpreta- tion without proof. First, / being some function of x and y, f and f" being functions of x only, and the differentials not enclosed in brack- ets being partial, it is shown that Fmust have the general form Whence 204 CALCULUS OF VARIATIONS. Therefore Hence if ^is to be a maximum or a minimum,/' must vanish for all values of x^ and U must be of the general form which, since the last integral is constant and might be written F{x), is not a general problem of variations. Thus in this case the maximum or minimum value of f/must be sought, if at all, by the differential calculus; and if the limiting values of X and y be fixed, U can have no maximum or minimum state. In both these cases F involves the first power only of y, and they are therefore examples of Exception 2, Art. 51. 158. We may now, before considering the next case, pre- sent the following general view of the treatment of the terms of the second order according to Jacobi. Assume the equation U=J Vdx, where V is any func- tion of X, J, y . . . . y'^\ and regard the limiting values of X, y, y . , . . y**- 1) as fixed. Then, as before, the solution must be obtained from the differential equation M — o, which will in general be of the order 27t. Hence its complete integral will involve 2n arbitrary constants, and may be written and this solution is rendered complete when the constants are so determined as to satisfy the conditions at the limits. 159. Next the terms of the second order must equal T r^f -J^ SMdydx. JACOBFS THEOREM. 205 For we have always But if we vary these coefficients, leaving dy, Sy' , etc., unvaried, we shall obtain the well-known form for the terms of the sec- ond order in dV\ namely, df ^ ^ dydy' ^ ^ ^ dy" ^ ^ ^ dy^^''- -^ Therefore it appears that the terms of the second order in d^ must in any case equal half of what would result from varying those of the first order, supposing Sy, <5>', etc., to undergo no change. They should not, however, be considered as really arising in this manner, as y, y' , etc., receive no second incre- ment. But when the limiting values of y, y\ etc., are fixed, the terms of the first order in (^^ become / MSydx^ so that those of the second order must equal -/ SMSydx. 160. It is evident that the reasoning of Art. 132 would be equally applicable whatever might be the order of the difler- ential equation M= o, and we shall therefore assume at once that SMsLud J^ ' dMdydx will vanish if for Sy we substitute the variations of c^, c^, etc., being, as before, entirely indepen- dent. Then Sy', Sy\ etc., will become — - or ?/, -— - or n", etc., dx dx the differentials being total with respect to x. 206 CALCULUS OF VARIATIONS. It will next be shown that SM can be made to assume the form dM ^ Ady A^ ^ Afiy' -\. eiQ ^^A^d^^\ After this the terms of the second order can be integrated I r^^ d' V by parts, until they finally take the form SU= -J t-t^c multiplied by the square of a certain function, analogous to that previously found. As the proof of the last two pomts is necessarily difficult, the general reader may, without serious loss, omit the re- mainder of this theorem, or may at least assume the truth of the two following lemmas, whose use will be at once evident. Lemma I. (61. SM C2ixv always be put under the form SM=A8y^^^ A,dy + etc + ^"L ^^ ^Z"*^- We shall, for convenience, abandon our former notation, and, adopting that of Prof. Jellett, write Whence SM—dN — d . ^_ ^_ dx dx dP d'^P Si\ —1 -\- etc ± — - — - . (2) dx ^ dx'^ ^ ^ JACOBVS THEOREM. 20J Jm p For take any term as 6 -— ^^ = dti'^), where t = P„^. Now if in Art. 9, we put / for j/, ^'' for/, etc., recollecting that /', t\ etc., are the total differential coefficients of t with respect to X, we shall, by reasoning precisely like that there employed, find that dt^'^'^ = -^-; so that it is evident that (2) has been correctly transformed. But • ,,.„ = f|..^ + ^.y + etc +p,<^/''>, and p _ dV ^ ~ ~d/^)' Therefore Hence m ^;ir"^ ^;i'"^ ( dy^'^'^dy dy^'^^dy . dW ' dy^'^^dy^'^^ Now consider some individual term of this series, as rim //2 77 ^m ^j^m ^j,{m) ^y{l) -y dx"^ -^ ^^^ d^V where / is not screater than 7n, and k = , ■■ , ,,, . Now if / ^ dy^^^dy^^) equal ;;/, this term is already under the required form ; but if / be less than m, there will certainly arise from the develop- d^Pi d^ ment of — i- a term of the form -— , >^ (5>("^), the sisfn of these dx^ dx'' 208 CALCULUS OF VARIATIONS. terms being like or unlike, according 2iS m — I is even or odd. That is, if SM be fully written out, it will be found that with the exception of those terms which are already under the re- quired form, all the others may be arranged in pairs, the type of which is the pair But by a theorem of the differential calculus, any pair of the form (4) can be arranged in a series of the form (See Note to Lemma I.) Whence it appears that all the terms in dj/can be ar- ranged as stated at the beginning of this lemma. Lemma IL 162. If A, A^, etc., be functions of x, implicit or expHcit, and u any quantity which will satisfy the equation 7 72 A2^+4- ^y + 4t ^y + etc. = o -, (o ax ax then if we write U^ u { Aut-^--^Aiuty^j-Alut)"^^tz. I , (2) Udx will always be integrable whatever be the value of /, the integral taking the form J Udx = B/ + ^ B/'+ etc., (3) where B^, B^, etc., are functions derived from A, A^, etc. JACOBrS THEOREM. 209 As the proof of this theorem belongs entirely to the in- tegral calculus, we follow the plan of Prof. Jellett, and append it in a note (see Note to Lemma II.). Case 2. 163. Next let U — J Vdx, where V is any function of X, y, y' and y'\ the limiting values of x, y and y' being fixed. Then, proceeding in the usual manner, the general solu- tion must be found from the differential equation M = o, where dV d dV d' dV ~ dy dx dy' "^ dx' dy"' ^^^ The complete integral of (i) will give 7 in the form y=f{^x, c,, c^, c„ c,) =/, (2) in which the four constants must be so determined as to satisfy the given values of y^, y^, y/, yj. But when these limiting values are fixed, we need not ex- press the terms of the second order in the usual way, which expression would be difficult to transform ; but we may write at once SU=lXyMSydx. . (3) We have now an invariable method of transforming SU, since we can always, according to Lemma I., put dM under the form 6M=Ady-\-^A,dyJr-^Afy\ and we shall now proceed to apply this lemma in order to determine the functions^, A^ and A^. 2IO CALCULUS OF VARLATIONS. [64-. For brevity of notation, let ayy^ ayy>, ay>y', ayy, ay^ and Uy^y denote the second differentials of V with regard re- spectively to y, y and y , y , y and y" , y' and y , and y" . Then, referring to the value of i^ in (i), and writing its variation in full, recollecting that the variation of the differential of any quantity equals the differential of the variation of that quan- tity, we have SM = Uyy dy + ayy, (5>'+ ayy dy" - — {ay,y dy + ay,y, (5>'+ a^y 6y") = ayydy - — ayy dy + — ayy 6/ + [- — ayy, Sy + ayy. dy' + (£. ayy dy + ayy dy') + g, a^y dy' - ^ a^y dy') "^ -"^ dx\ ^yy^-" ) ' dx' "^"^ ■" ' \dx + (£^,,,^ + ,..y) + g...y-^.3c^r), (4) where Now the first three terms of (4) are already in the required form, so that, setting these aside, we will consider the first couple. Here / — o, n= i, and there can be but one term resulting from this pair. Therefore, by equation (13), Note to Lemma I., the couple becomes ^c^^y^'\ or c^dy, or -£.:;,^K^y^ or ^ak,.dy, or -j- {- ayy).dy, (6) dx ax JACOB!' S THEOREM. 211 because a is always unity. Now consider the next couple. Here / = o, n= 2, and the number of terms which will re- sult is two. Hence, by (13), the pair becomes We also have by equation (14) of the same note, since a is al- ways one, and b is in this case two, and '''y+T/''y'='^'y+ii.^'^^y' ^ 2ayy6/. (7) dx^ dx In the last pair we have /= i, ?/ = i, and it becomes dx\dx ^ I dx\ dx -^ J ^ ^ Collecting results from the last members of (4), (6), (7) and (8), and arranging, we have dx Afiy' + i^A,,/, where dayy, d'ayy. or ttyy — a'yy, + a" yy* , ^.-^^.V + ^ + 2^..-> or — ay'y> + a ' y'y -\-2ayy; • A^ ^ayy. (9) 212 CALCULUS OF VARLATIONS, 165. We may now write and we know that if ii be an admissible value of Sy, u having the form given in Art. i6o, (^^can be rendered zero, and we infer, as in the first case, that there will be neither a maximum nor a minimum. But since the limiting values of y and y' are to remain fixed, we must, in order that ^y may equal u, or ku, be able to so determine .the constants dc^, dc^, etc., that both u and ic' shall vanish twice simultaneously at or within the lim- its of integration. In the former case we may change y into y-\- ku throughout the limits, while in the latter we make this change merely for the limits at which u and u' vanish, leaving y unvaried throughout the remainder of the integral. Also, since the variations of y, y' and y" must be infinitesimal, to make ^y equal ku, we must have u, u' and u" finite throughout the limits for which they are employed. 166. But suppose that the terms of the second order can- not be made to vanish by the use of u. Then if, as before, we put ut for 6y, (lo) will become zrri/ Itdx, (I I) in which we know, from Lemma II., that Idx is immediately integrable, giving f Idx = B/ + 4- ^"^"- (i2) '-^ dx ' 167.' Let* us next determine the functions B^ and B^. From (lo) and (i i) we have JACOBPS THEOREM. 21 3 ^« + -^^,«'+^^,^<" = o (13) ax ax' and I=^Aut + ^- ASut)'^ ^1 Alut)" \ u. (14) Whence, multiplying (13) by ut, and subtracting from (14), we have 1= u4-A,{uty + 7c-f^A,{uty - ut-^Ay - ut^-Ay dx dx dx dx = u [AXutyy + u {A,{utyY-ut{AX}' - ut{Ayy. (15) Now we know from Note to Lemma II. that all the terms in / which contain t undifferentiated must eventually cancel, so that we may neglect the last two terms in (15), and may also reject all others in / as they arise. We have then u\Ai2ityY^\2iAStityY - u'Aluty and {tity = ut'-\-ti't. Whence u\Aluty \ ' = {iiA.ut'y + {uA, 7i'ty - 21' a, uf - u'Aji't. Now the second and third terms of this equation can be united into one by Note to Lemma I., because here / = o and n ■=^ \. But as this term would certainly contain / undifferentiated, we need not perform the operation, but may reject them to- gether with the last, retaining only («M,o'. (16) Again, we have u\Aiuty'\"-=\uAi2ity'\"- 2 \u'Aiuty\'-^ti"Aiuty' and {Uty ^ 2a"-\-22c't'+lc"t\ 214 CALCULUS OF VARIATIONS, SO that u\ASut)"\"^ {uA,ut")"^2{uA,u'ty-\-(uAyt)"- 2{u'A,ut")' - ^{tt'A^ u't')'-2{ii'A^ u"ty+ 2i"A,ut"-\- 2u"A, u't'^ u"A, u"t. (17) Now set aside the first and fifth terms, which are already in- tegrable ; reject the last, and also the couple 6 and 8, because they could be united into one term, n being i, and that would contain / undifferentiated, because / is zero. Then there will remain two couples ; viz., terms 2 and 4, and 3 and 7. The first, since I— i and ?i—i, becomes \{2uA,2i'yfY, (18) In the last couple / = o, ;2 = 2, and it becomes {uA,uyt + 2{uAyty', and rejecting the first term, we have 2{uAyty. (19) Now collecting the terms from (16), (18), (19), and the first and fifth of (17), the result can be written thus: 1= {[u'A, - 4i/''A,+ 2uA,ii' + 2{uA,uyyY + {u'Ajy = {B,ty + {Bjr, (20) and this by immediate integration gives (12) ; and B, = tt'A, - 4u"A,^ 2uA^u" + 2{uA,tiy (21) and B, = ti'A,. (22) JACOBPS THEOREM. 215 168. We may now integrate (11) by parts, thus: SU^ 1 ritdx \^^AbJ-\- {Bjy \ - -» I Bj'+{Bj'y \ 2«^^o 2 L ^ ) 1 2 ( ) But If, therefore, we suppose u to be so taken as not to vanish at either hmit, t^ and t^ must vanish, and we shall have SU=- \£l' \bj'^ {B^ t")' } t'dx, (24) But we see at once that in this case the terms of the second order require still further transformation, as they are not yet in a quadratic form ; and to this we now proceed. (69. Let Va be such a quantity as will satisfy the differen- tial equation B,v'a-^{Bya)' = 0, (25) Then by putting iia for v' a, we have B,Ua-\-{B.u'a)' = o, (26) Assuming for the present that Va and consequently Ua can be determined, (24) can be still further transformed. For we see from that equation that if Ua were an admissible value of f 2l6 CALCULUS OF VARIATIONS. throughout the limits, ^^ would to the second order reduce to zero. But whatever be the value of t' , we may certainly represent it by Uata^ and (24) will then become ^U^ - i/J' j B^Uata^ \.BlUata)'^; | Uatadx I /'^i where I^dx is, as we shall show, immediately integrable by the note to Lemma II., its integral taking the form fl,dx=Cj'a. (28) (70. To find C^, multiply (26) by tiata and subtract from the value of /^ in (27). Then we shall have I, = Ua [BlUa taYy - Ua ta{B, U ^' , ' (29) But Ua \ BlUa taY V=\ tla Blu^ ta)']' - u' a B^tla t^)' and Whence Ua \ BlUa ta)' ]' = {Ua B, U^ t' a)' + {Ua B, u' ^ 4)' U d Jd^ 11(1 1 fi 21 d x5g U 0^ 1(1, Now since all the terms in /, which contain 4 undifferentiated must cancel, we reject the last term and also the couple 2 and 3, because, as ;/ = i, they could be united into one term which would, as / = o, contain ta undifferentiated. For the same reason the second term in (29) is rejected, and we have JACOBVS THEOREM. 21/ and where J I, dx = Q t'. C, = B, tc\, (30) 171. Resuming (27), dU can now be integrated by parts, thus : I /*^i dx 2^x0 = -'- {C,t\t} +1 [Cj'at^ ^\£'Cj'\dx. (31) The following equations will also hold : Sy , iiSy' — dyti' f ii6y' — dyu' 21, Now since u does not vanish at either limit, and dy and Sy' vanish at both, it is evident, from the above value of t' , that // and t^ will become zero, which will cause /« to vanish at the limits. Then putting for Ua its value v' a, and for C^^ the value obtained by referring to equations (30) (22), and (9), we have I r^^ [ t' y ^^ = Wxo ^^v^^'^^'a -7- ^^' (32) 172- We must now determine the form of the quantity Va, and for this purpose we must evidently solve (25). Now by comparing this equation with (12), we see that Va is what / must become in order to render / Idx zero ; that is, to ren- der / zero. But / = tiSM, which will at once appear if, in the final value of (^ J/ given in Art. 164, we write Sy = tit, Sy' — {tit)' and Sy'^ = {^i^Y, which will in no way restrict the values of the 2l8 CALCULUS OF VARLATLONS. variations. Hence, since u does not vanish, we must, when /is zero, have ^M zero. Now we already know that this condi- tion will be satisfied by making- = ar^-^hr^-\-cr,^-dr,, ' (33) and this condition can, since A, A^ and A^ are not in our power, be satisfied in no other way. For the integration of the equation M — o gives jr as a function of x and certain con- stants, the form of the function being determined, and the values of these constants only being undetermined. There- fore, since x does not receive any variation, any change which cannot be produced in 7 by varying the constants would cause some change in the form of the function, and hence y, when thus changed, could no longer satisf}^ the equation J/ =: o, which it must do in order that dM may vanish. This reason- ing is evidently applicable whatever be the order of M. Now it is evident that we can cause the second member of (33), which we know to represent the most general form of II, to assume various values for the same value of x by various determinations of the arbitrary constants a, b, etc. Let ti and V be any two such values, so that we may write u^a,r, + a^r^-\- a^ r, + a, r,, (34) v=.b,r,-\- b, r, -\-b,r, + b, r,. (35) But since Sy — tit, if we make / = — , (^J/ will become v, and the equation dM — o will be satisfied, as will also the equa- tion / = o. Moreover, this is the only solution ; since, by suit- ably determining the constants in v, - can be made to equal li JACOB!' S THEOREM. 219 any value of t which will render / zero, and therefore every value which will render / I dx zero. 173. But the value of v^ is not yet fully determined. For although, by substituting - for t, we shall render / zero what- ever be the system of arbitrary constants employed in v, we shall not, by such a substitution, necessarily satisfy (25). Be- cause when / vanishes independently of any particular value of V, J Idx is merely a constant. Hence all that we can say is that the relation Va— - will render the second member of u (25) a constant. Moreover, it is the only relation which will render it a constant, because it is the only value of t Avhich will cause / to vanish. Hence, since zero is a constant, if any real value of Va exist, it must be capable of being expressed in the form -; only t;ie eight constants, a^, etc., b^, etc., must u be so related as to satisfy (25). One of these relations will immediately appear. For, ex- amining (25), we see that it is a differential equation of the third order in Va ; and hence by integration we should obtain Va as a function involving not more than three perfectly arbi- trary constants of integration. If, however, we understand only by u and v any two quantities of the form given in (34) and (35) in which the eight constants are so related that -, when put for t, will satisfy (25), which relation must cause the V constants to be so combined that - may contam not more n than three arbitrary constants, then we may write v^ = -^ (36) 220 CALCULUS OF VARIATIONS. 1 74-. Although this relation between the constants was noticed by Jacobi, many subsequent writers have fallen into the error of supposing that they are entirely independent, and have thus rendered this portion of their explanation untrust- worthy. Among these writers is M. Delaunay, who was fol- lowed by Prof. Jellett. The latter, on page 95, makes a state- ment which would with our notation be equivalent to saying that whatever value of t will make / vanish, will also render / Idx zero, which is manifestly untrue. 175. We may now proceed to the final transformation of the value of (^t/ given in (32). We have, from (36), , uv — vu Vn. = ;: u u u Therefore t' uSy' — dyu' and / f Y_ {uv' - V7t') {u^y' - dyuj- (7iSy - dyti') {uv' - vit')' But {uv' — vuy = uv" — vu" y and (^^,Sy' — dyu')' = u^y" - Syu". Substituting these values in (32), reducing, and factoring with reference to (3j/, dy' and ^y" , we finally obtain [ uv' — vu' f JACOBrS THEOREM. 221 From this equation we see that to render U a maximum or a minimum, ayy must be of invariable sign, and should also remain finite throughout the. range of integration, and not vanish permanently. If these conditions be fulfilled, it is necessary also that the second factor of (37) should not per- manently vanish, and it ought also to remain always finite. The first condition will always be satisfied. For if in any case it were not, we would have SU=\£'6M6ydx = o; (38) and since every element of this integral must have the same sign as ayy, which is invariable, (38) can only be satisfied by making dM dy constant. But since Sy^ and dy,^ are zero, this constant must be zero also, which would render it necessary that SM should vanish. But this, as we have shown, would not happen unless u or ku be an admissible value of Sy ; and since, as explained in Art. 165, we assume in (37) that such is not the case, it is evident that the factor in question cannot perma- nently vanish. Hence we see that if Uy-y be of invariable sign, while SU cannot be made to vanish by the use of u or ku, as indicated in Art. 165, neither can it be made to vanish by any other mode of varying j/, y a,ndy'\ To satisfy the second condition it is necessary that the denominator in (37) shall not vanish, and that the coefficients of Sy and dy in the numerator shall both remain finite. That is, we must be able to so determine the constants that uv' — vu' may not vanish, while u, u\ u" , v, v' and v" must at the same time remain finite. But before we can examine these conditions, we must be able to express these coefficients of Sy, Sy' and Sy" as functions of x, and per- fectly arbitrary constants, and we shall next consider how this may be effected. (39) 222 CALCULUS OF VARIATIONS. 176. Now from (34) and (35) we have u = a,r, -\- a^r^ -\- a,r^ + (t,r„ ] u' = a,r/ + a,r,' + a,r/ + a,r/, V = b^r, + b^r^ + b^r^ + b,r,, v'^ b^r; J^b^r: + b,r: -\-b,r:, W'= b,r: + b,r:+b,r:+b,r:.] As we wish to substitute these quantities in the various parts of the second member of (37), we can avoid tedious multiplica- tions and exhibit the results more explicitly by the use of de- terminants. For (37) may evidently be written dU 1 T"^ 2t/a-o dx I r^^^ 2t/Xo LySy-Ly,S/ + Ly.Sy"\ where Ly» — Uy u Vs V Ly» (5j/, dy\ dy" u, u', u" v.. v\ v" u' , u' j dxy Ly LyI — u, u V, V (40) (41) Now for convenience we shall denote any determinant of the second order containing two ^'s and two ^'s by the numerical suffixes of its first element, and similarly determinants with respect to r, r' , etc., will be denoted by the numerical suffixes JACOBFS THEOREM. 223 of their first elements, together with the accents of r. Then, since u' , tt", v' and v" have the forms given in (39), while Ly is a determinant of these quantities, we can, by a well-known principle of the subject, at once exhibit Ly thus : Ly=:i2'i'2"^iyi'7;'-\- 14. iV + 23.2Y + 24.2r + 34'3V; (42) and in like manner we obtain Zj,^ = 12. i2''+i3.i3'^+i4- 14^^+23. 23''+24. 24^^+34. 34^ Zy/,= i2. 12^ + 13. 13^ + 14- 14' +23. 23^+24. 24' +34. 34^ (43) Hence if we regard the determinants 12, 13, 14. 23, 24, and 34 as new constants, we see that the eight constants in u and v have so combined as to leave but six in equation (40). If now we divide Z^, Ly> and Lyn by one of these constants, as 12, and denote the respective quotients hy My, My' zw^ Myn ,wq may, without altering the value of equation (40), substitute these quantities for Ly, Ly> and Lyn. Hence we require only to determme the forms of these quantities. But if we write a^'X b^l% c^X d='A e^^X{AA) 12 12 12 12 12 ^ ^ then My = 1^2'^+ a 1^3'^+ d I V+ ^2^3^^+ d2r+ e 3V, ' JZ,.= i2'^+^i3^^ + ^i4'' + ^23''+^24^' + r34^ I (45) Myn= 12' -\-a 13' + 3 14' +^23'+ ^24^ +^34^ ^ We have now but five constants to consider, and the last of these may be expressed in terms of the other four. For we have 12.34 + 23.14- 13.24 = 0, (46) 224 CALCULUS OF VARIATIONS. an equation which will be found upon trial to be identically true. Hence and 34 i_?i.H_ 13.24 _ 12"*" 12 12 12 12 e-\-cb e z= ad — be = a, b c, d ad (47) which value being substituted in (45) will give My, My> and Myi> as functions of four constants only. Our reasoning thus far would hold even were the eight constants which enter u and v entirely unrestricted. But since these constants must have such mutual relations as will satisfy equation (25), where we now know that Va is put for -, u the four remaining constants must also be subject to some restriction, or conditioning equation, which will enable us to express My, My> and My» as functions of not more than three perfectly arbitrary constants. But to determine this last rela- tion in any particular case it will be convenient to present equation (25) under another form, and this we now proceed to do. 177» Assume the equations Az + {A,z')' + {A,zT =f Au+{A,uy+{Ayy' =r and Then u/-zF=uAz-zAu+u{A, z')'—z[A,u')' -\-n{^A,z")" -z{A,u")"^i-^k. (48) JACOBrS THEOREM. 225 Now u(A^z')' = {uA.zy — u'A^z and zi^A, u'Y = {zAyy — z'A, u'. Whence i = A^(uz' — zu')'. (49) Also, u(^A,z")" =z {uAXY' - 2{u'A,z"Y-\-u''A,z"; and developing the remaining term in like manner, and sub- tracting, we have k = \A,{uz'' - ztOY' - 2\A,{u'z" - z'u")Y. (50) But since the second members of (49) and (50) are integrable once, if we add these equations, obtaining thereby the value of uf — zF, and then integrate, we shall have J \uf— zF\dx = A,{uz' — zic') — 2AJ^ti'z" — z'u") + lA,{uz--zinY. (51) Now put ^y for z, and let 2i be such a value of z or Sy as will render F zero. Then the second term will disappear from the first member of (51), and the remaining term will become / Idx; and we shall have f/dx = A,{udy - 6yu') - 2A,(u'dy' - d/?/') + \A,{uSy' - Syzc") Y = B,f + {B, fj. (52) t=^-l- But since t —^-^ and Va = -, we have only to chansre ^y into u u 226 CALCULUS OF VARIATIONS. V in order to cause / to become Va- Hence, finally, (25) may be written ASuv' - vu') - 2A,{u'v''' - v'u") + \AJ^uv" - vu")\' = o ; (53) and as we may divide by any constant, we may write, as the final conditioning equation, A, My. - 2A, My + {A, My)' = Q. ■ (54) It also appears by differentiation that Lyi = L'y» and L'y> =z Ly-{- uv'" — vu'" := Ly-\- Lx, where Lx is exhibited by determinants in the same manner as the other Z's. Hence, dividing these equations, as before, by the determinant constant 12, we have My. =r M'y., M'y. = My + M^, ) [ (55) M,= i2"'-\-ai3".'+di4''' + c23"' + d24'''+e34'''' ) It is evident, however, that in order to apply equations (54) and (55) to the reduction of the constants, we must deter- mine the particular forms which are assumed by A^, A^, r^, r^, r^ and r^, which cannot be done so long as the problem re- mains wholly general. 178. The following example is presented merely as a means of illustrating the preceding discussion. Problem XXXI. It is required to apply Jacobis Theorem to Prob. V. Here ay>y> ~ 2, so that we have next to consider whether the terms of the second order can be made to vanish by the J A GOBI'S THEOREM. 227 use of u or ku. Now the general solution, equation (6), Art. 42, may be written Hence we have the following equations : r^ — x\ r^ —x\ r^ = x, r, = i, ^/ =?>^\ r^ =^2x, r/ =1, rl = o, r/' = 6xy (0 r/' = 2, r/' = o, r/^ = o, ^2 =0, r, =0, r, u = a^x^ -\- a^x^ -{- a^x -\- a^, u' — la^ x"^ -\-2a^x-\- d. o. (2) (3) Now if the constants in u can be so taken that u and u' shall vanish twice or more, simultaneously, within the limits of in- tegration, the terms of the second order can be made to vanish by the use of u, and we have in general neither a maximum nor a minimum. Now if u and u' can satisfy these conditions, let x^ and x^ be two values of x for which they vanish Simultaneously. Then we must have < + ^^,' + ^^, + -'- = o, «1 *==+^^/+^^3 + -- = 0, ^1 x: + — ? ;ir, + —? = o, 3^1 3^1 3 I 2a^ . a^ ^3 + — -^3 H — -' = o. 3^1 3^1 (4) (5) (6) (7) 228 CALCULUS OF VARIATIONS. Subtracting (7) from (6), and (5) from (4), and dividing by ^%~ ^Tg, we have ■^. + -^3 + -7-0, (8) < + ^, ^s + < + " (^, + ^,) + -'- = O. (9) Substituting in (9) the value of --from (8), we have x; + -^2 -^3 + ^" - I {-^. + -^3) (-^2 +-^3) + -- = o 2X„X^ x: , a ^ + ?- Oo) 2 2 <^j Substituting in (6) the values of — ' and — ^ from (8) and (10), we have, after reducing, X^ — 2X^ X^ + X^ — O. Hence x^ and x^ cannot be different values of x^ and the terms of the second order cannot be made to vanish by the use of u . But since, as we have seen in Art. 175, these terms can be made to vanish by no other mode of varying j, we are sure of a minimum, unless, indeed, we cannot prevent My or Myf from becoming infinite, or My" from vanishing within the range of integration ; and these points we shall next consider. 179. Finding, by the use of equations (2), the values oi My, My, and My in equations (45), Art. 176, and also that of Mx in equations (55), Art. 177, we shall obtain Myu = — X* — 2ax^ — ibx'' — cx'^ — 2dx — e, My> = — 4x^ — 6ax^ — 6bx — 2cx — 2d, My — — 6x' — 6x — 2c, M^ = — 6x' — 6ax — 6b. (II) JACOBVS THEOREM. 229 Now since ayuy,, = 2, we see from equations (9), Art. 164, that A^—o and A„ = 2. Hence, in this case, equation (54), Art. 177, becomes - 4My + 2 J/V ^ o = - 2My + 2 J/^, as will appear from equations (55) of the same article. Equat- mg the values of My and M^, we have c = ^l?. Now taking the value of e from equation (47), Art. 176, and then substi- tuting in the first of equations (11) 3^ for c, we shall have, after changing signs, — My>^ = x' -{- 2ax^ + ddx"" + ^^^ -\- ad — 3<^'', (12) — My' = 4x^ -\- 6ax'^ 4" 1 2^^ + 2dy (13) — My = 6x' + 6ax + 6b. (14) It therefore at once appears that neither My^ nor My can be- come infinite so long as a^ b, d and x remain finite. We can also evidently choose these constants in such a manner that Myn shall not vanish within the limits of integration. For suppose, for example, that we make both a and d zero. Then to render the equation M ,t ^* 3 2 possible, we must have Hence if we assume b greater or less than this value can be- come within the hmits of integration, and also make a and d zero, we shall secure that Myi, will not vanish at all as we pass from x^ to ,r,; and therefore, as all the requisite conditions can be satisfied, we are in this case sure of a minimum. 230 CALCULUS OF VARIATIONS. (80. We have, then, the following general method of ap- plying the theorem of Jacobi in this case. First find whether ayuyu remains finite, does not vanish per- manently, and is of invariable sign throughout the range of integration ; because if these conditions be not fulfilled there is no need of any further investigation. But if they be satis- fied, next try whether dU can be made to vanish by th© use of u. For this purpose we write and a^ a^ a^ u' = r/ + --i r/ -t- -- r,' + -^ r/. a, a, a. Then if SU c?in be made to vanish by the use of u, the follow- ing equations must be possible : u^^ — o, u^ = 0, u^ = o, u^ = o, where neither x^ nor :r^ must fall without the limits of integra- tion. To determine the possibility of these equations we first eliminate between them the constants — , ~ and --, by which we shall arrive at an equation containing only x^, x^, and such constants as enter y in the equation of the curve represented by the solution. It may then happen, as in the preceding exam- ple, that we can determine the possibility of satisfying this equation within the limits of integration. Or, if necessary, we can, by using the values of y, y\ etc., obtained from the equation of the curve, eliminate all constants but numbers, thus securing a numerical equation between x^, x^, y^, j/g, j/, etc., which it must be possible to satisfy within the limits of integration. JACOBPS THEOREM. 23 1 If, then, it be possible to satisfy this equation, we infer, as in Case I, that we have neither a maximum nor a minimum. But if the hmiting values of u and u' cannot be made to vanish simultaneously, we may assume that we have a maximum or a minimum according as ay,>yn is negative or positive. This assumption will, however, be subject to any doubt arising from the possibility that we may not be able by any selection of constants to prevent My or My, from becoming infinite, or Myn from vanishing for some value of x within the limits of integration, thus rendering the corresponding ele- ment oi ^U infinite. To dispose of this doubt, we must, in the next place, actually find the quantities My, My, and My,,, and possibly Mx', as functions of x, and but three arbitrary con- stants, any constants which may enter r^, etc., not bemg reckoned. But this latter step, which will usually involve difficulty, may in general be omitted. 181. Some exceptions also occur in the treatment of this case which are similar to those mentioned under Case i (see Art. 157). We shall, however, merely indicate these excep- tions, the discovery of which appears to be due likewise to Spitzer. (See Todhunter's History of Variations, Art. 276.) Suppose, first, ay„y„ to become zero. Then it is shown that in order that^may become a maximum or a minimum, A^ must have respectively a positive or negative sign throughout the range of integration. Suppose, in the second place, that we have ay„yn zero, and also A^ zero, A and A^ having the values given in equations (9), Art. 164. Then it is shown that in order that U may become a maximum or a minimum, A must be respectively negative or positive throughout the range of mtegration. Moreover, in this case, as in Case i. Art. 157, we shall find that the equation J/= o will not be a differential equation in y, but merely an ordinary algebraic equation, and that there- fore y will, without integration, be determined as a function of 232 CALCULUS OF VARIATLONS. X, Hence, geometrically, there will be no solution unless the limiting values of y and y' happen to satisfy the equation of a particular curve or class of curves. Suppose, lastly, that ay^yf,^ A^ and A become severally zero. Then, as in Case 2, Art. 157, the equation U — J Vdx is capable of being integrated, and therefore the maximum or minimum state of U must, if at all, be found by the differential calculus ; and if the limiting values of x, y and y' be fixed, U will have neither a maximum nor a minimum state. It is evident that in all these cases V contains merely the first power of y" , and they are, therefore, like those in Art. 157, only examples of Exception 2, Art. 51. (82. As the most general case of Jacobi's Theorem is pre- cisely analogous to that already explained, and as it is rather of analytical than practical importance, we shall merely indicate the method of effecting the required transformation. Case 3. Let U = J Vdx, where Fis any function of x,y, y\ .... y^'^^. Then the general solution J/ = o will usually give y as a function of x and 2n arbitrary constants of integration, and these 2n arbitrary constants must be so determ.ined as to sat- isfy the conditions at the limits, where we shall always suppose the limiting values of x, y, y\ . . . . y^''^ - ^) to be assigned. Now, as before, since these conditions hold at the limits, and the terms of the first order must vanish, we may write and may then, by Lemma L, put ^J/ under the form 6M=Ady-{- [A,Sy +etc +[AnSj /(») JACOBVS THEOREM. 233 We shall also have, in this case, u = A,r,^ A, r, + etc: + ^sn^sw Hence, by changing (^J/ into ul, and integrating by parts with the aid of Note to Lemma II., we shall obtain a result which may be written - ^X^ 1 ^^^'+{bJ')\ etc. .... +[Bj''f~'' } /V^. Then, as formerly, putting Ua^a for /', and integrating again by parts, we have d[/ = jfC { ^^ ^'^ + (^^ ^'«) + ^^^ + (^^ ta^--'^^'"^^ \ t'adx. In this equation we may change t' a into ?/64, where ?/& = zv\, and wi) is a quantity which satisfies the differential equation C,zv\-\- [c,w\'^-\- etc + ((f^-^ftC^-DJ^''''^^ o. Making this change, and integrating by parts, as before, we have ^U=--J^;\Dj',^[Dj\)^^tc. ! \ (n - 3) ) Continuing this process n times, we shall evidently arrive at a result which may be written 2 ^^0 234 CALCULUS OF VARIATIONS. the positive or negative sign being used according as n is even or odd. Now it is evident, from the mode in which the integration is effected, that H must equal A^ ?/ u^a u\ . . . ., and An= ± — -1, the positive or negative sign being used according as n is even or odd, as will at once appear if w^e form the functions A, A^y etc., by Note to Lemma I. (83. Let us next consider the quantities zi, zia, Uh, etc. We have, by the same reasoning as that hitherto employed, u — a,r,-{- a^r^ + etc + ^su^sn, i^a — 'v' a> Va= -, v = b,r,-\- b^r^ + etc + b2nr2n li (0 But the 2n constants a and the 2n constants b are not entirely independent, but must be so related that Va may satisfy the equation B, v'a + [b, v"}I^ etc + (^,, vjf^^'^' '^= o ; (2) that is, Va, when put for t, must render / Idx zero. The following relations are also evidently true : / = 21 SM, /j = tiaj I dx, /j — tij) J /i dx, etc. (3) Now to determine the nature of Uh, we see from (i) that zvi, is a quantity which, being put for /«, will render y I^dxzQro\ that is, will render /j or ?/« j /^;t- zero, will render / Idx or udM zero, will render dM zero. But since, in / Idx, t' is replaced by Ua ta. JACOBTS THEOREM. 235 f ta — — 'j and in order to render that integral zero, the f in the value of ta just given must now be so restricted as to sat- isfy the equation B, t'+in, t'^+ etc +[b^ t(-)Y " '^ = o ; that is, it must render / Idx zero, or / zero, or m^M zero, or (SM zero. But always f = — , and we can make SM vanish only by making Sy equal to some quantity of the general form of u. Assume ^y = w, where w = c,r, + c,r, + etc + C2n ^2^. This will make ^M vanish ; and if the 271 constants c and the 2n constants a be suitably connected, t, which now equals — or Wa, will also satisfy the equation / Idx = o, which would not necessarily happen if these constants were entirely inde- pendent. We have now, as the value of /« which was required to render / Idjt^ zero, (?)• ^a — 1 — \t- - ul But it does not follow that every value of /« which will render J Idx zero will also render / /^ dx zero, and w^ must be such a value of /«• Still it is evident that wi^ can be of no other 236 CALCULUS OF VARIATIONS. general form than that just given for ta\ only, in addition to the relations already noticed between the constants, the con- stants in u, v and w must be so related that-; — -/may render It } /'• dx zero. In a similar manner we may determine Uc = z' c, but will then be obliged ultimately to introduce into dt/" another quan- tity of the form z-=d^r,-\-d^r^^ etc + ^2^ ^2^. Moreover, these four sets of constants will then be subjected to three more conditions ; six in all. For Zc must be so taken as to reduce to zero the following expressions : fl.dx, Jl,dx, fidx, SM; the last condition serving merely to introduce z, but imposing no restriction upon its constants. Thus it appears that each increase by unity of it will intro- duce into SU one more quantity like z, and that each such new quantity will require one more additional condition than did its predecessor, the first condition being introduced by the second of these quantities. Now T can always be found in terms of the preceding quantities. For we have ^ = — , ta = ~, h = i^, etc. (4) U Ma Uh Whence w^e see that by means of 7^ the final value of SU\N\Vi be made to involve dy, Sy\ .... dy^), which should evidently be the case. JACOBI'S THEOREM. 237 184. The analogy of the preceding cases would lead us to expect that when the reductions indicated in the last article are performed, c^^will assume a determinant form; and such is the fact. This subject, and indeed the whole theorem of Jacobi, has been most elaborately discussed by Otto Hesse in a paper which may be found in the 54th volume of Crelle's Mathematical Journal ior 1857, p. 227, and we are indebted to this author for much of the preceding discussion, and in par- ticular for that part which exhibits the relation between the constants and the manner in which they combine and reduce. We shall, however, here merely give some of his results. Let ti, V, w, . . . . X hQ n quantities which, being put for dy, will severally render 6M zqyo. Then we see from Art. 183 that dU will involve all these quantities. Let L = Sy, d/, .... r^J/(«) u, u\ .... ^^("•> X, x\ x^-^ Lyn u^ u , . . . . u in - 1) V, v\ -.X^-1) x.x\ Xin - 1) L being a determinant of the order 7i-^\, and Z^ a deter- minant of the order n. Then Hesse shows that dU will take the form \_ r^^ d'V I L dU dy^^)' \L ,yn dx. It is also evident, from Art. 183, that the number of the con- ditioning equations between the constants involved in u, v, . . . . X must be the sum 14-2 + 3-}- etc 4-/2—1, or n{n — i) 238 CALCULUS OF VARIATIONS. We may here collect a few of these conditioning equa- tions, the first arising from u^,, the next two from 2^5, and the last three from Uq. ^,^'a + etC +(^^^^Jr.)y^~'^ ^O^ C,w\^QtC -|-(|(f^,^,(n-1)y^"'^^0, / \ (71 - 1) ^,^^a+etC J^\B^^^^n)\^ .^o, f \(n-2) C, z\ + etc ^[Cn ^b^^ - i>j - o, / \(n-3) A-^'c + etC J^[DnZ,^n-2)\ ^o^ w z ^ a Wa — -, Za — -, U U where Some of these relations are more explicitly exhibited by Hesse, but, for a reason which will presently appear, it is unnecessary to go any further into this matter. Now _m Zc = — ,— » '^b = —r-, and Va = -' Hence it appears that ^5 is a differential expression of the first order, Zc of the second, etc. 186. The manner in which the constants enter <^6^ is similar to that in Case 2. For L may be written Ly dy+ Ly. dy + etc + Z^n dyin\ JACOBPS THEOREM. 239 where Ly, Ly>, etc., are themselves determinants of the nth. order. But if in any of these determinants we substitute the values of its constituents, we know that the determi- nant will become the sum of products of pairs of determi- nants, each product consisting of a determinant of the nth order in constants, multiplied by a determinant of the same order in the rs and their differential coefficients, there being as many such products as there are combinations of 2n num- bers, taken n in a, set, no two determinants, whether variable or constant, being the same. This is, however, as far as we can go. For to show, in general, how these determinant constants combine so that Ly, Ly>, .... Lyr. may be expressed as functions of x and en- tirely independent constants, is a problem which has not yet been solved. Now in order that no element of (5' ^maybe- come infinite, we must be able to so determine the arbitrary constants that Ly^ shall not vanish, and that none of the quan- tities Ly, Ly,y .... Ly^ may become infinite within the range of integration. But the above defect will prevent us from determining whether or not these conditions can be fulfilled, since it will prevent us from obtaining these quantities as ex- plicit functions of x and entirely independent constants. 186. After a general discussion,Hesse considers successively the cases in which ?/ is i, 2 and 3. In the latter case, the con- stants will enter Ly, Ly>, .... Lyn, in the form of twenty deter- minant constants of the third order, and the conditioning equa- tions will be three in number. Moreover, between these twenty determinants there subsist thirty identical equations analogous to equation (46), Art. 176. Now by division, as before, Ly, Ly>, .... Lyr. become My, My,, .... J/^n, and these constants may be reduced to nineteen. Then the three conditioning equations should enable us to reduce them to sixteen, and finally the thirty identical equations are of such a character as to enable us to eliminate but ten more determinants. 240 CALCULUS OF VARIATLONS. Thus it will appear that there remain not more than six irre- ducible constants. Hesse does not say that these constants are yet perfectly independent, and the author is not prepared to say more than that they appear to be so. For a further discussion of this subject the reader is referred to the paper in question. 187. We see then, in general, that in order that U may have a maximum or a minimum state, it is, in the first place, necessary that or ^^n^n shall remain finite, not vanish per- manently, and be of invariable sign throughout the limits which we wish to consider. This principle, however, is not due to Jacobi, it having been enunciated by Legendre as early as the year 1786. Still, the method of discriminating maxima and minima given by Legendrp and Lagrange was defective, because it gave no means of determining whether some element of c^f/ might not become infinite, as it always employed certain functions which could not be determined. (See Todhunter's History of Variations, Arts. 5, 199.) If the conditions with regard to ay^yr. indicate a maximum or a minimum, we must, in the next place, determine whether d^can be made to vanish by the use of u, since, if it can, there is no need of Jacobi's transformation, and we infer at once that U has neither a maximum nor a minimum state. To make d^thus vanish we must be able to satisfy the equations U^ — O, u,' = 0, etc., ^^(n-l)_o, u^ — 0, u,' = 0, etc., ^^(n-l) — Q^ where neither x^ nor x^ must fall without the limits of inte- gration. Geometrically, we may regard j in any proposed solution as the ordinate of a curve whose extremities are the points ;i:o, Jo and ^i, j\. Then the proposed value off will render U JACOBPS THEOREM. 24 1 neither a maximum nor a minimum, if it be possible, by mak- ing infinitesimal changes inj,y, . . . .y^), to draw another curve meeting the first at the points x^, y^ and x^, y^, and having at these points the same values of y, .... y"- - 1>, and also satisfy- ing the equation M ^ o. Now although, when the limiting values of x, y, y\ . . . . y{n-\) are assigned, all the constants which enter the equation of the curve which satisfies all the conditions of the question are determined, yet as this determination is not alway abso- lute, allowing us a choice of two or more values, there will in general be more than one such curve,^as in Prob. VII., where two catenaries can often be drawn, both satisfying the condi- tions of the question. Now if such limits be found, in passing along- one of these curves, as to render it and another curve coincident between these limits — that is, if the equation of this curve have one or more pairs of equal roots — dU to the second order can be made to vanish, and we infer that U has neither a maximum nor a minimum state. If we can assure ourselves that SU will not vanish, then we must, in the third place, determine Ly, Ly>, .... Z^.», in order to ascertain whether or not all the elements oi dU remain finite. But this point has been already fully treated, and we have seen that this determination cannot always be effected. When this is the case, the theorem of Jacobi is practically subject to the same defect as existed in the method of Legendre and Lagrange. It will appear, however, that by determining the function tc, which is always possible when the complete inte- gral of the equation M=o can be obtained, and sometimes when it cannot, we may frequently be able to infer that U has neither a maximum nor a minimum state, even when dy^yn is always finite and of invariable sign ; and this inference could not be drawn from the above-named method. 188. From the cases in which n is i and 2, we might natu- rally expect that some exceptions to the theorem of Jacobi 242 CALCULUS OF VARLATLONS. would arise wnen n is greater than 2, particularly if ayr^yx. should happen to become zero throughout the range of inte- gration ; and such appears to be the fact. For Spitzer has examined also the case in which n is 3, and has shown that certain forms of Fgive rise to exceptions. We subjoin from Todhunter's History, Art. 278, the following four forms of F, which the reader may examine for himself : v=A^,y,y',y")^y"'fa{x,y,y',y"), V = /(^, y, y')+y"fa {x, y, y')+ \ h{x, y, y, y") \ ', V = Ax,y)+y'fa{x,y)+ \f^{x, y, y')\'^ \ fc{x, y, y', /')]', V=yAx)+\fa{x, y)V+ \Mx, y, y')V+\fc{.x, y, y', y")}', where /, /«, fb and fc are any functions whatever. Hesse does not mention the existence of any exceptional cases, although he had seen the discussion by Spitzer. It will be observed that in applying Jacobi's Theorem we have always regarded the limiting values of x, y, /,.. . y^-*) as fixed, thus rendering the discussion somewhat restricted. But the solution of the more general problem, that in which these limiting values are also variable, if it be at all possible, has up to the present time baffled the skill of those who have attempted it. 189. Before closing this section we must mention one point with regard to the terms of the second order not strictly con- nected with the theorem of Jacobi. We have already seen that the simplicity of the form in which these terms appear is often dependent upon our choice of the independent variable, and it may therefore be well to consider particularly how the terms in 6U, derived by regarding x and y successively as the independent variable, are connected, and why they are not identical. CHANGE OF INDEPENDENT VARIABLE. 243 Assume the equation u^jydx, (I) where Fis any function of x,y,y' , .... y^), the limiting values of X, /, y, . . . .y"'-i) being fixed. Then, since both x and y are implicitly, at least, involved in U, we may regard y as some function of x, and may therefore suppose it the ordinate of some primitive curve for the abscissa x. Now by varying U we must pass to some derived curve, and let Y or y -{- dy be- come the ordinate of this curve for the same abscissa x. Next taking/ as the independent variable, and expressing U in terms of y, dy, x and its differential coefficients with respect to j, equation (i) may be written V.^Jyy',dy, (2) the limiting values of y, x, x\ etc., being also fixed, where dr x' — -^, etc. Moreover, ^and U. will be identical when the dy relations between x and y in (i) and (2) are the same ; that is, when y is an ordinate of the same primitive curve in both for the same abscissa. Varying (i) and (2) and transforming the terms of the first order, observing that the limiting values are all fixed, we have \-^^'^^s.y^yd^^\s.yd^+s^!'d-^' (3) I' UA =Sy'- ^^ + J ,CS dy + £ Tdy, (4) where brackets denote the entire increment which ^and U^ receive by variation, the integrals following M denoting re- spectively the terms of the second order and all those of a higher order, and those following N having a similar significa- 244 CALCULUS OF VARIATIONS. tion. Now supposing U and U^ identical, the first members of (3) and (4) will become equal if in (2) we so vary x as to obtain the same derived curve as we did from (i) by varying y ; and this requires dx to have such a value that y may be the ordinate of the derived curve for the abscissa x -|- ^x. Hence, by tracing along the derived curve from the point whose co- ordinates are x and Y to that whose co-ordinates are x -{- dx and y, we see that . y= V+ Ydx + - Y'Sx' + etc. ; and putting for V its value y + ^y, we find dy = — y'dx -\- w, (5) S.= -% , (6) y where w contains, only terms of an order higher than the first. Now, since the second members of (3) and (4) are absolutely equal, the terms of the first order in these two members can- not .differ by any term of the first order. Hence, and from (6), observing that dx = -^, we have, to the first order, MSydx= NSxdy=: -Ndydx; so that N = — M. Still, denoting by a the terms of the first order, and by b those of the second, in (3), and by c and d the corresponding terms in (4), we cannot say that a and c are absolutely equal, but they cannot differ by more than some term of the second order, and they will in general differ by such a term. In like manner, a -\- b and c -{- d cannot differ by any term of the second order, although they may differ by some term of the third, and therefore b and d may, and in general will, differ by a term of the second order. CHANGE OF INDEPENDENT VARIABLE. 245 190. Now if ^is to be a maximum or a minimum, and we express it successively as in (i) and (2),- then a and c must each vanish, because M and N vanish, and we may then find that d is much more simple than b ; and as these terms must now be equal as far as the second order, because a and c have become zero, we conclude that b must contain an expression which adds nothing of the second order to its value, and that this, by the second method, becomes involved in c, thus leav- ing b in the simpler form d. We know, moreover, that M and N will be entirely inde- pendent of the conditions which may be required to hold at the Hmits, so that the relation N — — M must hold whether the hmiting values of x, y, /, etc., be assigned or not. Now if the limiting values of x be fixed, while those of y are vari- able, then if we change the independent variable to y, we may, by regarding the limiting values of y as fixed and those of x as variable, pass to the same derived curve as by the first method. But the abscissae of the extreme points will now be x^ -{- dx^ and x^ -\- Sx^, whereas they are required to be x^ and x^ merely. Hence to render \_^U^ and [^^] equal, we must subtract from the former the increment which U would re- ceive in virtue of the change in the limiting values of x ; and we know that to the first order this increment is V^Sx, - V,Sx,. (7) Now when [SU] and \SU^ have been made equal, as just explained, it is easy to find what must be the values of the co- efficients of Sx^, dx^, 6x^', etc. For let ;/ be 2, and x the inde- pendent variable. Then, by equation (5), Art. 36, the terms at the upper limit will be (A - Q/W. + QA\'' (8) But from (5) we have, to the first order, ," Sx: ; (5) 248 CALCULUS OF VARLATIONS. and if by the aid of (2) we express (5) in terms of (4), it will become identical with (4). In like manner we might treat the terms at the lower limit, only adding V^dx^. We also have M = 2f\ N = = 5 d x'" dy x" + d' X'' '^ dfx" = 2X'^ x" 20;ir'V^' x^' • + Zox'" x" = ■ - 2f\ as will appear by consulting the value of f^ given in (2), so that here also N — — M. Section IX. DISCONTINUOUS SOLUTIONS. (92. We now enter upon a portion of our subject which is of comparatively recent development, but is nevertheless of the highest analytical importance. But some general view of the nature of discontinuous solutions having been presented in Art. 103, we shall, without further explanation, proceed at once to the consideration of an example to which the develop- ment of the subject is chiefly due. Problem XXXII. // is required to determine the form of the surface of revolu- tion which shall m,eet the axis of revolution at two fixed points^ have a given area, and enclose a maximum solid ; the two fixed points being so taken as to render a sphere inadmissible. ' It will readily appear that this is merely Prob. XVI. with an additional restriction upon the limits, which can in no way DISCONTINUOUS SOLUTIONS. 249 affect the general value of U. . Hence, as there, we shall, assum- ing X as the axis of revolution, have and the limits being fixed, we must have, to the first order, Now if, as usual, we make (J Evanish, we must have M=2y-\-2a Vi +/' - A__^^ML^ = O. (2) To integrate this equation, write 2ydy-\- 2a Vi -\- y'"^ dy — 2ay' . d yy Now f2a VTT7' dy = 2a \^VT7'y - f -j=y, ^/ ' and _f2a^^_2ag^r_^l^^ Hence, by reduction, (2) gives 250 CALCULUS OF VARIATIONS. and since the curve must meet the axis of x^ c vanishes, and we must have / H — =r^=rz —y \yA ^ C = o. (as This is equation (4), Prob. XVI. ; and if we make dU z^ro, we are necessarily led to this equation. (93. But the equation at which we have now arrived ad- mits of two solutions, j/ = o and y -\ — = o. The first, however, cannot hold throughout the entire range of integra- tion, since the surface generated is to be a given finite area, while the second will, as we have already seen, lead to a sphere, having its centre on the axis of x^ and is therefore ex- cluded by the conditions of the problem. We are naturally led next to inquire whether the solution sought might not be obtained by combining in some manner the preceding solutions. Thus, in the figure, let A and B be the two fixed points. Then if the given surface be less than TtAB'^y we may suppose the generating curve to be AECB\ and when the surface exceeds tcAB^, we may suppose the genera- trix to be AFDB. Under this supposition we know that the revolution of the semicircles will render the given integral, when taken from A to C, or from A to Z>, a maximum, while the line CB or DB may be considered as generating a cylinder whose .diameter is infinitesimal, and whose surface and volume are so likewise. Here, however, a new difficulty presents itself. For if in M DISCONTINUOUS SOLUTIONS. 25 I we substitute zero for y, observing that y will be zero also, we shall obtain M = 2a\ so that it appears that y = o is not a solution of the equation J/ == o, and that, therefore, if this latter equation is to hold throughout the entire range of integration, this solution must be abandoned also. The fact is, however, that we cannot reject the solution y =z o because it does not satisfy the equation J/ = o, since the question now involves a principle of variations which we have not hitherto considered ; and this we next proceed to explain. 194-. In former problems we have been obliged to con- sider Sy as capable of having either sign, and therefore, when 6U was developed into a series, and the terms of the first order transformed in the usual manner, we were compelled to equate M, and also the coefficients of Sy^, 6y^, Sy^\ etc., sev- erally to zero, as the only means of preventing the terms of the first order from exceeding the sum of all the others, and thus rendering the sign of (^6^ positive or negative at pleasure. But by referring to Art. 99 we see that the conditions of this problem prevent y from becoming negative, and hence when y is zero — that is, when the primitive curve coincides with the axis of x — we can give y positive increments only. To determine in the most general manner what effect this restriction would produce when applied to the present prob- lem, let us suppose that U and V retain the same form as before, but that the limiting values of x and y become vari- able, so that we shall have +J^^'Mdydx + etc., (5) where brackets denote the entire variation of U, the etc. the terms of an order higher than the first, and M has the form 252 CALCULUS OF VARIATIONS. given in (2). If now we suppose y to become zero throughout the range of integration, (5) will become \PU^ =fJ\adydx-\-Qtc.=jJ'— 2r Sy dx -{- ttc, (6) where r is a positive constant, since it appears, by referring to Art. 99, that 2a must be essentially negative. Since the proposed solution j/ = o does not reduce the terms of the first order in [dW] to zero, but merely to a single term, it is plain that this term will exceed the sum of all the fol- lowing terms, and hence that its sign will control that of \_^U\ But because dy is now necessarily positive, the sign of this controlling term is no longer in our power, but is essentially negative, thus rendering [^W] a negative quantity of the first order. Hence, if we wish to render C/a maximum or a mini- mum, not as compared with all consecutive states of ^ which can be produced by varying y and y', but with such only as can be obtained by making Sy invariably positive or negative, we see that the solution j/ = o will, in the former case, render U a maximum, but in the latter a minimum. We may call such maxima and minima conditional maxima or mi7iima. Now as the sign of (5^6^ will depend upon that of the term of the first order, we have in this case nothing to do with the terms of the second order, and thus the problem is much sim- plified, unless, indeed, these terms should happen to become infinite, which would, as before, throw doubt upon the whole solution. But this will not occur in the present case. We see, then, that in this case, by restricting Sy to one sign, we render it unnecessary that the proposed solution should reduce M to zero, and also remove the necessity of an examination of the terms of the second order. Neither was it necessary that M should become a constant, but merely that it should be finite and of invariable sign. But should J/ change its sign, we could, in the same manner as has already been ex- DISCONTINUOUS SOLUTIONS. 253 plained for terms of the second order, cause the term of the first order, and consequently [dU\to assume either sign at our pleasure. Simple as is the foregoing principle of restricting Sy to one sign, it appears to have been first introduced into the calculus of variations by Prof. Tod hunter, in the Philosophical Magazine for June, 1866. It will, however, when somewhat more ex- tended, afford the basis for some important investigations, and will also serve to explain some points which have hitherto been the source of difficulty to the student in this department of analysis. (95, In applying this principle to the present problem, let us first suppose the given surface to be less than that of a sphere having AB a.s its diameter, and let the abscissa of A be x^, that of B, x^, and that of C, x^. Then suppose the inte- gral to be divided into two parts, the first extending from A to C, and the second from C to B; so that we may write ^^ly^^+iy''^^- (7) Now, supposing M to be zero throughout the first integral, its variation will reduce to ^a; and putting jj/ = o, y = co , observing that 2a — — 2r = — R, R being the radius of the sphere, this term will also vanish. If we vary this portion of the integral only, while leaving the rectilinear portion unvaried, we shall, theoretically, be obliged to examine the sign of the terms of the second order ; and we have already seen that this investigation is not altogether sat- isfactory. Still, as it is well known apart from the calculus of variations that the sphere is the solid of maximum volume for 254 CALCULUS OF VARIATIONS. a given surface, we may assume that f/ will in this case become a maximum ; that is, that \SU^ will become a small negative quantity of the second order. Now throughout the second integral we have M=2a= —2r= — R, and the variation of this integral becomes / — R6y dx, which must be negative, since Sy is invariably positive ; and thus in this case the whole integral U must become a maximum. It should be observed that while the values of x\ and y^ are the same for both parts of the solution, those of y for the same point differ. Thus for the circle y^ is infinite, while for the rectilinear part j/ is zero, and we shall be obliged sometimes to observe this and similar distinctions with great care. When the given surface exceeds that of a sphere described upon AB as a diameter, let x^ be the abscissa of D. Then we may consider U as consisting of two integrals, the first extend- ing from A to D, and the second from D to B, and we may still write This mode of considering the integral may seem erroneous, inasmuch as it will compel us to regard x as doubling upon itself at D, and x^, therefore, as greater than x^. But it is to be observed that we assume that U and SU are continuous integrals — that is, that they are capable of being expressed by one definite integral — and this requires that x shall be uninter- rupted. Adopting for the present this view of the subject, we see, as before, that if we vary the arc ^Z^ only, (J ^ must become a negative quantity of the second order, and that if we vary the line DB also, we shall have 6U = J^ — Rdydx, DISCONTINUOUS SOLUTIONS. 255 which is negative, as before. In this case we in reality reckon twice the volume generated by 6y along DB, when we pass to the derived solid. We may, however, construct the two solutions as in the subjoined figure, in which case we shall be obliged to con- sider U as consisting of three integrals, and we have therefore adopted the other construction as being more easily explained. B i> 196, We have already shown, in Art. loi, that when the dis- continuous solution is necessary, that necessity arises from the fact that the conditions which require the surface to be given, and the two terminal points on the axis of x to be also assigned, have been so fulfilled as to render them incompatible with the general solution. Now we shall find, as we proceed, that dis- continuous solutions generally, if not always, arise from some incompatibility in the conditions of the problem, and that the conflicting* conditions are imposed sometimes consciously, that is explicitly, and sometimes unconsciously, that is im- plicitly. The present problem would, of itself, afford an example of the former kind, but it is in reality onl}^ a comple- tion of the discussion suggested by Prob. XVI., and there the discontinuity was of the latter kind, arising from conditions incidentally imposed, the effects of which were not foreseen. 197. We will next consider an example which will serve to extend our theoretical knowledge, and to prepare us for the discussion of more important questions. ^5^ CALCULUS OF VARIATIONS. Problem XXXIII. Let U ^^ I {y""^ — 2y)dx = / Vdx^ and let it be required to maximize or minimize U, the limiting values of x being fixed^ those of y being zero, and it being also required that a certain fixed point, whose co-ordinates are x^ and y^, shall not fall without the required curve. This is, in fact, merely a restricted form of Prob. V., and we have J/— 2/^— 2. (l) Now if it be possible to draw between the fixed points on the axis oi x a curve satisfying the equation M = o, and also enclosing the point x^, y^, there will be no difficulty, and we shall have a minimum as in Prob. V. If the point should happen to fall upon the curve, dy^ could not be made negative, but this would not affect the problem, since the curve would render ^a minimum for all admissible variations oi y andy^ Suppose, however, that no curve satisfying the equation M = o can be drawn so as to enclose or pass through the point x^,y^. Still, as the sign of ^y is wholly unrestricted, except at the points x^ and x^, and also possibly at the point x^,y^, if the curve pass through that point, we feel sure that M must vanish throughout the entire integral U. We are therefore naturally led to inquire whether the solution might not be furnished by drawing from x^ and x^ severally an arc of a curve satisfying the equation M = o, the two arcs meet- ing and not excluding the point x^, y^ ; and such a solution we now proceed to consider. 198. Let the arcs meet at the point x^,y^; x^ and ^fj being less than x^. Then U may be written U=£ydx+fydr, (2) DISCONTINUOUS SOLUTIONS. 257 where Fhas the same form as before. But although the two arcs satisfy the same differential equation M — o, still the con- stants which enter their equations cannot be identical ; other- wise they would form one and the same curve, which is contrary to our present supposition. Hence by making M zero in (i), and integrating, the general equation of the two arcs may be written X 24 24 (3) If now to the first order we take the variation of each inte- gral in U separately, and transform it in the usual manner, observing that dy^ and Sy^ vanish, and that the parts which remain under the sign of integration must also vanish, because M is zero throughout U, we shall obtain SU=- 2y:"Sy, + 2 Y:"S Y, + 2 Y:'S Y: - 2y:'Sy: ^2y-Sy:-2Y:'SY:. (4) Since all the variations in this equation are of unrestricted sign, SU vnw^t vanish ; and also if (4) be expressed so as to in- volve only variations which are entirely independent, the co- efficients of these variations must severally vanish. Now if we assume that x^ does not vary, Sy^ and dY^ are the same quantity. Moreover, from (3), we have y:":^x,^ec, and Y:" = x,^eg. (5) Whence -2y:"Sy,-\-2Y:"dY,= i2{g-c)Sy^', (6) and since Sy^ is certainly an independent variation, c and g must be equal. 25 o CALCULUS OF VARLATLONS. Now consider the t^xvcis 2y^'Sy^ — 2Y^'SY^. To make these terms vanish we may have jg^' = Y^' , and also ^y^— ^ Y^\ or Iz^ = o and Y/^ = o. Under the first supposition y^^ and Fg' must mean the same thing, otherwise their variations would not be necessarily equal. If now we equate jj/g'^ to Y^'\y/ to Y/, and/s to Fg, taking the values of these quantities found by differentiating (3), we easily discover that if c and £• are equal, c^ = g-^, c\ = g^ and ^3 = ^3> which is, as has been shown, not admissible in this case. But suppose jg^^ = o and Fg^^ = o. Then y^ and Fg^ need not mean the same thing, and we have only c ^^ g and c^ ■= g^. But there still remain in (4) the terms 2Y^'SYI — 2y^" Sy^\ and to make these vanish we must have Y^" = o and yj' = o. Take the origin midway between the points x^ and :i\, and let x^ = e and ;t:o = — ^. Equating the values of y^^' and Y/\ as found from (3), we have - + ^£-^+2g = --6ce + 2c,. Whence, since c ^ g and c^ = g^, c and g are zero. The equation Y^' = o now becomes — h 2^1 = o, and the 2 equation Fg^^ — o gives —^-\-2g^=z o, impossible equations un- less e"" and x^^ be equal, which they cannot be since x^ was taken numerically less than x^. Hence we must abandon this solution, since it will neither cause SW to the first order to vanish nor to have an invariable sign. (99. There remains but one supposition, which is that the arcs be drawn as before, but meet at the point x^, y^ ; and this, which we shall find to be the correct solution, we next proceed to consider. It is plain that C/and also ^C/ will have the same form as before, except that the suffix 3 w^ill be changed into 2. But it DISCONTINUOUS SOLUTIONS 259 will not now be necessary to make all the terms in ^ Evanish, because 6y^ and S K, are the same quantity, and dy^ is neces- sarily positive. Now, as before, the terms involving these two variations will become 12 {g— c) Sj/^, and it will be neces- sary to make the remainder of (5^ C/ vanish, because it contains only variations of unrestricted sign. We have then to examine whether we can make these terms vanish ; and if so, what will then be the sign oi g — c. To make the terms involving 6y^' and d F/ vanish, we have, as before, . F/'=j/' and F;=j/,'. (7) The first equation being necessary, and the second also neces- sary unless F/^ = o and j//' = o, a case which we shall subse- quently consider. To make the terms involving d F/ and Sy^' vanish, we have, as before, F/^ = o and y/' = o. (8) We have also, from the other conditions of the question, 7o = o, Y, = o, V,=f,; (9) and these equations, together with (3), which still holds, will be sufficient for our purpose. 200. Take the origin as before, and also denote x^ by a, and jTg by d. Then finding, from (3), the successive differen- tials of y and F, (7), (8), and (9) give, by substitution. — + 6ac-\- 2c, = -- + 6^^+ 2g,, (10) 26o CALCULUS OF VARIATIONS. 3 3 ^+6^^ + 2^x=0, ^--6ce^2c, = o\ (II) 24 24 ^^^ -j- iTj^''^ — ^2^ -|- ^3 = o ; (12) 24 24 +^'«'+e{c-\-g\ (20) Now from the second of equations (10), we obtain g,-c, = 2a{c, - g) + ia\c - g) = 6ae{g+ c) - 3a\g- c) = |^, (22) the second member being obtained by (20), and the third by (21) and (16). Hence, by substitution, (19) becomes a'e — ^' , '^V — 5^' _ 2be 262 CALCULUS OF VARIATIONS. Whence Now we will so estimate y as to make b positive. Then, since e exceeds a, to make g positive, we must have or 2-^x5^'- ' 12 24 4 24* (23) But in Art. 42 we showed that the general equation of the single curve meeting x at the points — e and + ^j J'/ and j^' not being fixed, is -^""24 4~ 24* Hence, if the members of (23) were equal, b, or y^, would be an ordinate of a single curve drawn from the pomt x^ to ;r,, and satisfying all the other conditions of the problem. But since y^ or b is m this case too great to be made the ordinate of any such single curve, the conditions of (23) are fulfilled, and g must be positive. Therefore, in this case, ^C/ reduces to \2{g— c)dy^, which, because Sy^ cannot become negative, is positive, so that these arcs furnish the minimum solution required. 201. We have still to consider the casein which j/' = o and IV' = o. Here the second of equations (10) is not neces- sarily true, but the first may be replaced by the two equations - -\- 6ag -\- 2g = o and - + 6ac -\- 2c^ = o. (24) 2 2 DISCONTINUOUS SOLUTIONS. 263 Then from (11) and (24) we readily deduce a-\- e J • a — e p- = ' — and r = . ^ 12 12 Hence d^^ becomes in this case dU^ I2{g— c) dy^ ^ — 2e (5>,, which, as ^y^ is still positive, must be negative. Thus it appears that this solution will render U a maxi- mum. Now in the present case g is necessarily negative, and we have seen that when Y^' = y^' and F/ = 7/, g is neces- sarily positive. Hence, when we satisfy the first of these equations by making Y^' and y^' severally vanish, the second cannot hold true. We see, then, that the minimum solution consists of two arcs which satisfy the equation M = o and meet at the point x^, y^, so as to have there no abrupt change of direction, and to make their radii of curvature at that point equal and finite. 202. In closing this discussion we must observe, first, that when we propose to make the two arcs meet at the point ;r„, y^, it is by no means the same as if we had been required to draw each arc so as always to pass through the two fixed points. For then y^ would have no variation, and we would treat each curve separately by the well-known rules of varia- tions. In the second place, the terms maxima and minima are here also used in the technical sense already explained, and we must be careful not to say that the present solution gives the least value of W. For B being the point x^, y.^, we can, by a construction hke that of the figure, make U as small as we please. 264 CALCULUS OF VARLATIONS. All that follows from the preceding discussion is, that if we draw two arcs as required by the solution, and then, regarding this curved line as a primitive, pass to any other curved line which can be derived from the first by infinitesi- mal variations of y and y" , the variation of y^ being positive, U will be thereby increased by a quantity of the first order. If we make dy^ zero, the proposed solution will reduce the terms of the first order to zero, and we shall be obliged as usual to appeal to those of the second order, which will be dU= r^dy"^ dx + r^d Y"'' dx, which, being positive, will render U in this case also a mini- mum. , 203. We may now with profit consider partially the gen- eral theory of discontinuous solutions. Suppose we wish to determine the relations between x and y which will maximize or minimize the expression U — J^ Vdx, V being, as usual, any function of x, y, y', etc. Then, after the usual transformation, we may write SU=L,— L,-\r r^M dy dx, where L, and L^ have the well-known form of the terms at the limits. Now if no restriction be imposed upon ^y, we know that M must vanish throughout the entire range of integra- tion, and likewise Z^ and L^ must vanish. But suppose the problem be such that ^y must always be positive or always negative ; then it may not be necessary to make M vanish, provided it be of invariable sign, and pro- vided, also, that the terms at the limits either vanish or be- come of the same sign as the unintegrated part; in which case DISCONTINUOUS SOLUTIONS. 265 (J C/ will become a quantity of the first order, and there will be no need of examining the terms of the second order. Suppose, next, that U is such that it may naturally be divided into a number of integrals, say n, the first extending from x^ to x^, the second from x^ to x^, etc., the last extending from x^n_^ to x^ ; and suppose dy is of invariable sign through- out one or more of the intervals into which x is divided, but is unrestricted throughout the others. Then M must vanish throughout the latter ; but if throughout each of the former M be of invariable sign, and if the sign of Mdy be the same throughout each, M need not vanish provided certain con- ditions can be secured at the limits, and we shall have a dis- continuous solution, made up of curves satisfying different differential equations. But when the sign of ^y becomes necessarily invariable throughout any interval, we shall find that this restriction results from the fact that there is throughout that interval some boundary which the required curve is forbidden to pass ; and in order that the sign of Sy may be made invariable by this boundary, the required curve must, throughout that inter- val, coincide with it. It will, therefore, readily appear that whenever any portion of the required solution does not satisfy the equation M — o/\t can consist of nothing but a portion of some boundary, the nature of which will be generally known. Thus in the case of a sphere, this boundary, although not ex- plicitly assigned, is easily seen to be the axis of x, the implicit condition that y is not to become negative making this the boundary below which y cannot pass. If, however, the sign of ^y be restricted for some point or points only, as in the preceding problem, the equation M = o must hold throughout U, although the equations of the arcs for different intervals may differ widely in the values of the constants which they contain. It will, we think, now be evident that, in general, when a discontinuous solution presents itself, it will be made up in 266 CALCULUS OF VARLATLONS. one of these three ways : first, some combination of arcs satis- fying the equation J/ = o ; second, some boundary or certain boundaries ; third, some combination of this boundary or these boundaries, with arcs satisfying the equation M =^ o. 204. Let us now consider the integrated part of dU, when U is divided as explained above. As the different portions of the discontinuous solution meet at the points whose abscissae are x^, x^, etc., jk will have the same value for two curves meeting at those points, but the values of y , y^, etc., for two curves at their points of meeting may differ widely. To recognize this distinction, we employ the suffixes 2 and 3 to denote quantities both of which corre- spond to x^, but belong to different curves meeting at the point x^,y^, and we divide x into x^, x^, x^, etc., the last being x^, the suffixes 3, 5, etc., being reserved for the second of the two quantities corresponding to x^, x^, etc. Now performing the integration for each integral sepa- rately, the first gives L^ — Z^, the second L^ — Zg, etc., so that the entire integrated part of (^^ becomes Zj — Zo + Z. — Zg + Z, — Z54- etc. +Z2n_2—Z2^_„ or Z. (i) Now if all the variations involved in Z be of unrestricted sign, it must vanish ; and also if Z be transformed so as to con- tain independent variations only, the coefficients of these vari- ations must severally vanish. But suppose some of the varia- tions involved in Z to be of restricted sign. Then, the other terms having vanished as before, it may not be necessary to make these terms vanish also. For if these restricted varia- tions be related, suppose them to have been reduced to inde- pendent variations, and let H, Z, K^ etc., be the several pro- ducts of each variation and its coefficient. Then we can reduce any of the quantities H, Z, etc., to zero by making its variation factor vanish. If, therefore, these quantities be all of Uke sign, that of Z is determined ; but if, on the contrary, they be not, Z can be made positive or negative according as DISCONTINUOUS SOLUTIONS. 267 we reduce to zero the negative or positive quantities. But suppose i/, /, etc., to be of like sign, making that of L the same, and that this sign does not conflict with that of M 6y in the unintegrated part. Then- dU becomes a quantity of the first order, having a fixed sign, and we need not examine the terms of the second order. But if H, /, etc., be of unHke sign, or if their sign, when the same, conflict with that of Mdy, L must vanish altogether. In equation (i) we have assumed that no portion of the axis of X is to be counted twice, as in Prob. XXXI I., where the sphere extends beyond x^, because such cases will seldom occur. When, however, they do arise, Z, although differing somewhat in form from (i), can be readily found by integrat- ing each portion separately, as before ; and then all the condi- tions which we have just explained will hold true for this case also. Problem XXXIV. 205, It is required to determine what will be the solution of Prob. VI I. when the two fixed points are so taken that no catenary ca7i be drawn between them having its directrix on the axis of x. Here U lyv.^-y'^dx^Sydx (I) and Now it is natural to inquire, first, whether any restrictions have, either expUcitly-or implicitly, been imposed upon the sign of Sy^ in virtue of which the equation J/ = o need not hold throughout U. For if not, the solution can consist of noth- ing that will not satisfy this equation. Now Vdx in (i) is the 268 CALCULUS OF VARIATLONS. value of any element of the generated surface divided by 27r, and it does not seem reasonable to suppose that this surface can ever become negative. Hence, since we take Vi-^-y'"^ positively, it would appear that y cannot be negative ; that is, that dy, along the axis of x, must be positive. We infer, then, that should the solution contain anything which does not satisfy the equation M — o, it can only be some portion of the axis of x, and that such portions will be likely to occur. 206. Let us next examine the equation i)f= o to see what can be obtained from this source. This equation will give / vY+y^ dy = vr+y-^y -f-y^. ^/ ^'''+y'-':7=k^^ + fy^-: yy Vi-\-y" y Vi+y Whence c. y _ Vi +y' (3) This is the same as equation (2), Art. 59, so that this is the only condition which can be obtained from the equation M=o. Suppose now that we digress from the method of solution pursued in Art. 59, and make c zero. Then (3) will give either jj/ — o or y =00 , and these two solutions, although neither can be employed alone, can be combined.. For let A and B of the figure be the two fixed points, CD being the axis of x. Then the discontinuous solution proposed will be the broken line A CDB. DISCONTINUOUS SOLUTIONS. 269 Thus in this case, as in Prob. XXXI I., the solution y = o, which arises from the same conditions in both, is suggested as one solution of the equation J/ = o, which it does not, how- ever, in either case satisfy. But this suggestion was not necessary, as this solution was anticipated by the reasoning of the preceding article, which would be equally applicable to Prob. XXXII. 207. We now proceed to show that the proposed solution will minimize U. As we cannot treat infinite quantities by the methods of variations, we shall, to avoid their occurrence, transform to polar co-ordinates. Take some point within the figure as the pole, the initial line being parallel to CD. Let V, the angle between r, the radius vector, and this initial, be estimated in the direction A CDB, and let k be the distance of the pole from CD. Then any element of the generating curve will be Vr" ^r'"^ dv, and its distance from CD will be k —r sin V. Then, ds being an element of the surface. ds = 27t{k — r sin v) Vr^ -\- ^ dv. Whence U=S^yk-rsxnv)^^^^^^dv=£ydv (4) and = X'' I ~" ^^' + ^'' sin vSr-^zrdr-^ zr'Sr' [ dv. (5) 270 CALCULUS OF VARIATIONS. But since the proposed solution cannot be represented by the same equation throughout, (3) and (4) must, without in any manner changing their form, be written as three integrals — that is, three times with different limits — the first portion, A C, extending from v^ to v^ ; the second, CD, from v^ to v^ ; and the third, DB, from v^ to v^. Then, transforming 6U in the usual manner, we have 6U= {zr'Sr)- {zr'Sr),-^ {zr'Sr\- {zr'dr\-\- {zr'dr),- {sr'dr\ -f fj^' I _ Vr' + r'' sinv + zr — -^ zr' \ drdv. (6) Now the suffixes 2 and 3 relate to C as being on the two lines AC znd CD, and the same is true of the suffixes 4 and 5, so that r, = r,, v^ = v„ r, = r„ v, = v„ Sr, = dr„ Sr, = Sr, ; while r/ and r^' differ, as do also r/ and r/. Now dr^ and 6r^ are zero, the points A and B being fixed ; and although the other suffixed variations need not vanish, still at the points C and D, for either line, we have k — r sin v = o\ so that all the integrated terms in (6) disappear, and we have left only the integral, which must be considered as divided into three parts, as just explained. 208. We may now write d U =fJ^MSr dv -^fJ'Mdr dv -{-fJ^MSr dv, (7) where M=. — Vr' + r'' sin V + zr - -^ zr\ (8) dv ^ ^ Now consider first the second integral. Along CD we have y^ — r sin ^ = o, so that M — — X^r" + r"" sin v, a negative quantity of invariable sign. But along this Hue dr is always DISCONTINUOUS SOLUTIONS. 2/1 negative, so that every element of this integral becomes a small positive quantity of the first order. Let us now examine the sign of the first integral. Along AC WG have r cos v a constant, say c, so that we find r sin V c sin v cos V cos'^ V * COS^ V zr = r = = ^-, Vr 4- r ' ^ cos Z^ COS V kr cos^ 7/ r^ sin z^ cos^ ^ c c r cos z/ = k cos V — r cos 27 sin v =^ k cos v — c sin ^'. Whence J/ = ^— 4- /^ cos V — c sin ^' cos z^ ' dv . , • \ sin e; ) \k cos V — c sm z') y cos 2^ ) c sin ^' • , 7 cos v cos' v = o, 2 dv and similarly, N will also vanish along DB^ because there we shall have r cos v ■= — c. Thus, finally, since ^r and ^r' are zero along CD, the terms of the second order reduce to the second integral in (9), which, as we have already seen, can never become negative. 210. It is plain that the discontinuous solution which we have just examined exists even when the fixed points are so taken that a catenary is admissible. But to determine in this case which of the minima gives to U the smaller value is a problem of the differential and integral calculus solely, and for this purpose we have the following formulse. For the discon- tinuous, s being the entire surface. For the continuous, let PT be any line tangent to the catenary at P, and meeting the axis of x at T, the abscissa of which is x^, and let 5 denote the surface generated by PT, while s denotes that generated by the por- tion of the catenary between P and its lowest point. Then, 274 CALCULUS OF VA^UATLONS. regarding x^ as positive or negative according as P and T are on the same or opposite sides of the axis of y., when it passes through the lowest point, we have in which a is the well-known constant of the equation of the catenary, and can be calculated approximately when the co- ordinates of the fixed points are given. It will be found, how- ever, that sometimes the continuous, and sometimes the dis- continuous solution will generate the smaller surface. ^ 211. We have seen by the reasoning of Art. 5 that the cal- culus of variations is not theoretically bound to furnish all possible solutions ; and since two may exist in the present problem, it is natural to inquire whether there may not be another, which will render the surface less than does either of those which we have considered. We reply that, while theoretically such might be the case, still no such solution has ever been discovered, and there would seem to be little doubt that one of the two already examined will always give the least, as well as a minimum value of U. In fact, we are now beginning, and shall continue, to verify the remarks of Art. 14, and to show that, although subject to some restrictions which would seem to greatly limit its power, the calculus of variations is in reality capable of furnishing nearly all the solutions pertaining to the maxima and minima states of irreducible integrals. We shall find, moreover, that these solutions will generally in some way present themselves as solutions of the equation M—o, although they may in reality not satisfy that equation at all. DISCONTINUOUS SOLUTIONS. 2/5 Problem XXXV. 212. A projectile which is acted upon by gravity alone is to start from one fixed point and to pass through another. It is re- quired to determine the nature of its path, so that the action 7nay be a minimum.'^ Assume the origin at the starting-point, and estimate x vertically downward, and let the initial velocity be s/2ga. Then we know that the velocity of the projected particle at any point of its path will be ^2ga(xT\- a). Hence u =X" n^ +«)(!+/') dx =£' vdx. (I) Whence, in the usual way, we obtain So that X ^ a — c and, by integration, y — c^— ±2 Vc{x -\-a — c). (3) Since X and y are simultaneously zero at the starting-point, we have — c^= ±2 Vc{a — c) and (3) becomes y±2 Vc{a — c) = ±2 Vc(x+ a — c). (4) * See Todhunter's Researches, Chap. VIII. 2/6 CALCULUS OF VARLATLONS. If c(a — c) be positive, (4) represents two parabolas, and we must now consider whether these parabolas can be made to pass through the second point. We have y,±2 Vc{a -c)= ±2 \/c{x, + a —c\ (5) so that, squaring, we obtain Whence ( J,' - ¥x^" = i6y,'c(a - c). Hence it appears that c{a — c) can never become negative, and (5) will therefore contain no imaginary quantity, unless c be- come imaginary. But the last equation may be written thus: i6c\x: + j^) - ^cy^ix, + 2a) +y: = o ; or, dividing by the coefficient of ^^ i^ may be written c'-2Pc+Q^o. (6) From (6) we may obtain the values of c, which, since P and Q are positive, will be real so long as Q does not exceed P^ the two roots being equal when P' = Q. Now the condi- tion P' > or = Q gives, by reduction, y,' < or =4a{x,-\-a). (7) Hence we see that if the first member of (7) exceed the sec- ond, c in (5) can have no real value ; if the members become equal, c can have but one value ; and if the first member be- come less than the second, c can have two real values. Now it is evident that for any given values of x^, y^ and a, but one of the forms of (5) can be true for the same value of c, and that therefore we can have, passing through the two DISCONTINUOUS SOIUTIONS. 2jy fixed points, as many parabolas as there are real values for c. But (7), when its members are equal, is itself the equation of a parabola, which may be called the limiting parabola. For we see that if the second point lie without this parabola, it cannot be joined to the first by any parabola which will sat- isfy all the conditions of the question, so that the solution, if one exists, must be discontinuous. If the point be on this parabola, one parabola only can be drawn ; w^hile if the point be within the limiting parabola, two parabolas can be drawn. Of course when the values of ^,, j, and a are fully given, we can determine the one or two equations involved in (5), so that they may be without ambiguity of sign. But when two values of c exist, we cannot determine which must be taken, unless we fix the angle which the projectile in starting makes with the horizontal, two angles being admissible. 213. Let us now examine the terms of the second order. We have Now when x decreases algebraically — that is, when the projec- tile is ascending — we must regard the velocity as negative. But then ds is also negative, so that, both radicals in (8) be- coming negative, SU wiW be positive. When x increases — that is, when the projectile is descending — both radicals become positive, so that SU'is positive. If, then, the arc of the parabola with which we are con- cerned does not include the vertex, we undoubtedly have a minimum ; but if we are required to reach or pass beyond the vertex, then, since y at that point becomes infinite, our conclu- sion that we shall have still a minimum cannot be regarded as altogether trustworthy, and we shall be obliged to resort to another method of investigation. 278 CALCULUS OF VARLATIONS. 214. Let us now assume the horizontal as the axis of x, estimating y vertically downward, and taking the origin at the vertical distance a above the first fixed point. Then we shall have , u=iywTr)dx=fydx. (9) Whence, by formula (C), Art. 56, we have Hence so that Whence x= ±2C,Vy-C, + C, y=C^ + ^—-^. (10) Differentiating (10), we have / = ^'. (n) Now (10) is the equation of a parabola when the directrix is taken as the axis of x, the origin being assumed at pleasure, and C^ is the abscissa, while C^ is the ordinate of the vertex, 46^1 being twice the parameter, or 2p. For making y' zero in (i i), we have x = Q, and then (10) gives, for the same point of the curve, y = C,. Now, changing the origin to the point Q, C^, we shall obtain, after interchanging the variables x and y, y" = a^C^x = 2px. Hence we see that the distance of the directrix above the starting-point is always numerically equal to a, or the height due to the initial velocity. DISCONTINUOUS SOLUTIONS. 279 Now we know that the focus of any parabolic path de- scribed by a projectile moving from ^ to ^ must be at the intersection of two circular arcs, described with the same radius a from the two points respectively as centres. But if a be so assumed that these circles cannot touch, there can be no continuous solution, and the point B will be without the limiting parabola. If the circles touch, one parabola can be drawn, having its focus upon the line AB, the point B being then upon the limiting parabola ; while if the circles intersect, there will be two parabolic paths along which the particle may move, the first having its focus below, and the second above the line AB^ the point B being in this case within the limiting parabola. 215. It will be seen that by changing the independent variable we avoid any infinite value of y, and we will now pro- ceed to show that when the parabolic arc has its focus below the line AB, the action becomes a minimum, but that when the focus is upon or above AB, the action is not a minimum. Employing Jacobi's method, we have, from (9), or -^ — dr "" virryy (13) which is always positive, and remains finite throughout the range of integration, so that we shall have a minimum if we can take u so that it shall not vanish within the same range, and that u^ may remain finite. From (10) we have the value of y' being taken from (11). Therefore the most general value of u is u=i- /' - Ly\ and u' = - 2//' - Ly\ (15) 28o CALCULUS OF VARIATIONS. Now because y' and y" remain finite, u' will not become infi- nite ; and to make u vanish, we must have L=^,-y=-H, (i6) and we shall have a minimum if the range of H over all real values be only partial, but none if it be complete. 216. In order that H may range over all real values, it must certainly touch zero and infinity. The first condition requires 7' to become ±1, and is fulfilled at both extremities of the latus rectum, and there only. The second requires y' to become either zero or infinity, the latter condition being never fulfilled, and the former at the vertex only. Now let y^' and y^ be the values of y' at the extremities of any focal chord. Then, because the tangents to the parabola at these extremities meet at right angles upon the directrix, we must have y' = „ and hence we shall find that y:---r or H,^y:-~ or H,. )\ y^ Therefore as H in this case starts with a certain value, changes sign by passing through infinity at the vertex, and returns to its initial value, its range must be at least complete, and we have not a minimum. If the arc were still greater, the range of //"would be more than complete. Now H can return to its initial value but once, although it may pass that value. When the initial value is zero, this is evident, since, as we have seen, there is but one other point at which H can be zero. When the initial value is not zero, // must change sign twice before returning to its initial value, and four times before returning to it a second time, and this latter is impossible, since there are but three points at which H can change sign at all. DISCONTINUOUS SOLUTIONS. 28 1 Since, then, the values of H at the extremities of any focal chord are equal, they will be equal nowhere else, and the range of H is then just complete. If, therefore, the arc in question be less than that subtended by a focal chord, the range of H is not complete, and the action becomes a minimum. In other words, we see that the action will not be a minimum unless the second fixed point be so situ- ated that tw^o parabolic arcs are admissible, and then for that path only which has its focus below the line AB. 217- Since there can be no continuous solution when the second fixed point lies on or without the limiting parabola, we next inquire whether there ma}^ not be some discontinuous solution or solutions in these cases. We first ask, then, whether we have unconsciously imposed any boundary along which the sign of Sy is fettered ; because, if not, the solution can, at least so far as discoverable by the cal- culus of variations, consist only of some combination of lines satisfying the equation M — o. But we see from (lo) that y = o is such a boundary, since to make y negative would render the velocity imaginary, and with the notation of (i) this boundary is given by the equation x-\- a=^o. 218. Let us next see what can be obtained from the funda- mental equations given by the two methods previously em- ployed. These are because there is no escape from these, if we make M vanish in each case. The first of these equations is satisfied by y == o and ^ = o, and alsoy — oo , because in the latter case we ob- tain X -\- a^zc. Passing for the present the question of combining y = o and y == 00 , it is suggested that our solution may consist, m 282 CALCULUS OF VARIATLONS. part, of some line parallel to the axis of y. But because y' would here become infinite, we cannot, while keeping the ver- tical as the axis of x, investigate the variation of C/ along this line. But the second of equations (17), in which the axis of x is horizontal, offers the same solutions as the first, since it is satisfied by j' = 00 , 6^1 = o, or by y' = o, which gives y = C^j which is the same condition as was before expressed by X -\- a = o. With this change of the independent variable, we can ex- amine the condition which we were before unable to investi- gate ; namely, whether the solution may be composed, in part, of some horizontal line. Now if this horizontal be any other than the boundary / = o, it must, since along it ^y is of unrestricted sign, satisfy the equation M = o. But this equation, when C/"has the form given in (10), becomes M = 2 Vj dx ^\^ y' (18) But when we puty = o and y = C^, we have M=^ - — _: and 2 Vl, as this does not vanish, this solution must be abandoned. 2(9. As the horizontal line j/ = o is not yet known to be excluded, since it need not satisfy the equation M=^o, and as y' z=i 00 was also suggested as a solution, it remains to consider whether the solution may not be found by combining this hori- DISCONTINUOUS SOLUTIONS. 283 zontal with the verticals through the two fixed points, as in the figure, where the path of the projectile is supposed to be ACDB. Of course a particle could not move from A to B along this broken line, because its velocity along CD would become zero. But we can draw a curve indefinitely near to A CDS along which the velocity will not become exactly zero, and then we shall find that the action along this curve will be greater than that along the discontinuous path. To determine whether the line ACDB is the path of mini- mum action, we shall, on account of the infinite value of y' , need some other method of investigation, and we might try transforming to polar co-ordinates. Still an analytical demon- stration will not here be necessary. For let AC and At be equal in length, and let them be divided into the same number of equal and infinitesimal parts ; and let PQ and pq be a cor- responding pair, so that AP will equal Ap. Then because P is vertically higher than /, the velocity at P will be less than that at/, and the action through PQ less than that through/^. Hence it appears that the entire action through ^(T is less than that through Ac, In like manner we show that the action through BD is less than that through Bd. Now the action along CD is zero, while that along cd is not ; so that it is cer- tain that the action along the primitive A CDB is less than that along the derivative AcdB^ even if we do not vary the bound- ary CD, and much more so if we vary that line. 220. It is easy to see that the discontinuous solution which we have obtained is admissible even when the parabolic path also renders the action a minimum. When the second fixed point lies on or without the limiting parabola, the discontinu- ous solution, being the only one which presents itself, undoubt- edly renders the action the least possible, as well as a minimum. When both minima are admissible, we shall find that some- times the one and sometimes the other will give the smaller 284 CALCULUS OF VARIATIONS. minimum ; and there can be little doubt this smaller minimum is in every case the least possible value also of the action. The comparison of the two minima, when they exist, must be effected by the ordinary calculus, but we subjoin, without proof, the necessary formulae. (See Todhunter's Researches, Art. 173.) Let g be the force of gravity, r^ and r, the radii vectores of the two fixed points, C the length of the chord joining these points, and w the action. Then for the parabolic path, accord- ing as it subtends less or more than two right angles at the focus, we shall have or w = :^i(^ + ^ + 0«+ ('-0+'-. - Cf\- (19) For the discontinuous solution the action is that due to pass- ing along the verticals only, and is ^=^(..+..). (20) 221. The principles which have been previously explained regarding the origin and nature of discontinuous solutions are equally applicable when polar co-ordinates are employed, and we shall find in thi-s case also that they are generally in some manner presented as a solution of the equation J/ = o, although they may not, and need not always, really satisfy that equation at all. Let us now briefly consider a problem of this kind. DISCONTINUOUS SOLUTIONS. 285 Problem XXXVI. It is required to determine whether there be any disco7itinnoiis solution involved in Prob. XXII. We have seen, Art. 123, that when the second fixed point lies without a certain limiting ellipse, no elliptic arc, satisfy- ing all the conditions of the question, can be drawn between it and the first fixed point ; and that even when it is situated on the limiting ellipse, and there can be one eUipse drawn, it does not render the action a minimum. It appears, then, that if there be any solution in these cases, it must be discontinuous ; and the analogy of the last problem would lead us to expect, what is indeed the fact, that even when a continuous solution exists, a still smaller value of the action is in some cases given by a certain discontinuous solution. 222. Now the fundamental equation of this problem is equation (8), Wr"" where W^ i/--i, (2) r a ^ ' and we cannot help arriving at this equation if we make M vanish. But if in (i) we make c zero, that equation will be satisfied by / = ^ or W— o. The first would indicate that we might employ some portion of the radius vector drawn to one or both the fixed points, or of these radii produced. To interpret the second we have W' = --- = o, r a so that I I - = — , r = 2a. r 2a 286 CALCULUS OF VARIATLONS. That is, it is suggested that a portion of the solution might consist of a circular arc described from the centre of force with a radius 2a. Let O be the centre of force, A and ^the two fixed points. Then the discontinuous solution which is proposed is the path A CDB, where CD is the portion of the above-named circular arc intercepted between OA and OB produced. But before considering whether the proposed solution does render the action a minimum, we inquire whether any bound- ary exists along which the sign of Sr is fettered, and which need not therefore satisfy the equation J/ = o. Now the value of the velocity v' , equation (4), Art. 121, is \/^l--l=WVf, (3) where / is the intensity of the attracting force at a unit's dis- tance. When, therefore, VV vanishes and r becomes 2^, v' becomes zero ; and when we make r greater than 2a, v' be- comes imaginary. Hence the arc CD is itself such a boundary, unconsciously imposed, and along it ^r must be negative and the action zero. 223. Owing to the infinite value of /, we cannot deter- mine, by adhering to polar co-ordinates, whether the pro- posed solution Avill render the action a minimum or not, and the natural mode of procedure would be to express the value of dU\n rectangular co-ordinates, by which we could escape infinite values. But this will not be necessary, because, by reasoning precisely similar to that employed in Art. 219, it will appear that the action through A C and DB must be less than that through any curve of the same length which can be derived by the method of variations, and the arc CD cannot be reached by curves which do not exceed these lines. Then as the action is zero along CD^ it is evident that the discon- DISCONTINUOUS SOLUTIONS. 28/ tinuous path A CDB will render the action less than would any other path which could be derived from- it by the calculus of variations. Problem XXXVII. 224-. A steamer is to pass from one port to another on a stream whose current flows always in the same direction, Jier speed beiitg dependent solely upon the angle which Jier course makes with the directio7i of the current, together with certain constant quanti- ties. It is required to determine the fo^m of her path, so that the passage may be made in the shortest time possible. Assume the course of the current as the axis of x, and esti- mate X in the direction of its flow. Also let v be the velocity, t the time, and ds an element of the required path. Then since v depends, in some fixed manner, upon constants and the angle between the path and the axis of x, we may write V = F{y'), and ds VI -\-y'\lx , , y J dt — - — ^-^- — /( J ) dx = fix, V r Hence the expression to be minimized is ^= / fdx, where t/ Xq it is evident that /can become any function whatever oi y'. Now we have already seen. Art. 56, that the solution of this problem is given always by a straight line, and there is no escape from this conclusion so long as we make M vanish. For dx dy dx so that if M be zero, we cannot help obtaining f — c\ and to satisfy this equation, y' must certainly be a constant, which w^ill lead to a right line as the only possible solution. But since the required line is in this case to pass through two fixed 288 CALCULUS OF VARIATLONS. points, we seem at first to be restricted to a single course for all possible conditions, whereas a little reflection will serve to show us that we could easily impose such conditions as would enable us to shorten the time of passage by pursuing a path not always coinciding with the straight line joining the two points. It appears, however, upon examination, that the equation M—o must hold throughout the entire course, as we cannot find that any boundary has been in any way imposed along which Sy or dy' will be of restricted sign. We feel certain, therefore, that no solution can be obtained, at least by the cal- culus of variations, except a right line, or one composed of right lines. But since f is a constant, suppose that constant to become zero. Then if the equation f'-=o furnish more than one real value of y\ we may have two or more lines meeting at finite angles. For the terms free from the sign of integration, which are f^'^y^ — fj^y^ + etc., would vanish, because f would vanish for either of the meeting lines, although the values oi y' for the two lines might differ. We shall, however, illustrate this problem by considering some particular cases. 225. 1st. Let a be the angle between the path and the axis of X, which is not to exceed — , and suppose the velocity v to vary as cos a = = . Then in this case U be- sec a 4/i._Ly2 comes U=£\i ^y')dx =fj^'fdx, giving f ^ J- = 2/. Now y' must have the same value throughout the integral, because if it change value* at any point x^, j/^, we shall have, as already explained, without the integral sign, after trans- forming the term of the first order in the usual way, f:^y.-f:^y.. or 2{^y:dy,-y:dy^, or 2(j// -X^Jo, DISCONTINUOUS SOLUTIONS. 289 which must vanish, since Sy^ may have either sign. Hence, in this case, the minimum time will be gained by following the right line joining the two points ; and because only one value of y is admissible, we infer that this path gives also the least value of /, t being certainly a minimum, since the term of the second order is / Sy'^dx. In this case, then, there is no dis- continuity, but we now pass to an example in which it occurs. 226. 2nd. Let 2^4 so that /=-?+f <.) where b is some constant. Then proceeding as usual with the integral U= J fdx, we obtain /'=/(/'- 1) =^- (2) Whence we also find P-/"-3/'-i. (3) Now if we solve (2) without restriction, we shall obtain a straight line, which must of course pass through the two fixed points, and we will first examine whether this continuous solution will always render the time of passage a minimum. Now since the term of the second order in 6U is -jy'^y"'^-^ 2^^o 290 CALCULUS OF VARLATIONS. we shall have a maximum or a minimum according as f" is negative or positive. Hence, from (3), observing that tan^ — = — , we see that when the angle is less than — , the time is 63 6 a maximum ; but that if the ports were so situated that the line joining them must make with the axis of x an angle greater than 30°, the time will become a minimum. 227. Now when we have shown t to be in any particular case a maximum or a minimum, it does not follow that we have obtained its greatest or least value, since some discon- tinuous solution may give a greater maximum or a smaller minimum. Now if there be any discontinuous solution, it must cause f or c in (2) to retain the same value throughout U, otherwise there would arise terms of the form {f^' —fj)^y^, which would not vanish. Any values, then, of y' which will satisfy the equation /' = c, in which we may give to c any value we please, only retaining the same throughout U, may be combined into one solution, provided this combination will enable us to pass from one fixed point to the other, and pro- vided also that the various parts of the combination do not render the terms of the second order of variable or conflict- ing sign. Suppose, in the present case, we make c zero. Then we obtain, as the roots of (2), / = o, / — i, / = — i. But the last two values of / render f in (3) positive, while the first renders it negative, and cannot, therefore, enter any solu- tion with the other two, as the sign of the terms of the second order would then be in our power. It is evident that a vessel could pass from one point to any other by a suitable combina- tion of tacks, making with the axis of x angles whose tangents are either -|- i or — i ; and as the integral has the same value, whatever be the number of these tacks, because /is the same whether y be + i or — i, we obtain in all cases one path along which the time of passage will be a minimum. DISCONTINUOUS SOLUTIONS 29 1 To determine, when two minima exist, Avhether the quicker passage can be made by following the path composed of tacks or a continuous line, is not a problem of variations, but of algebra only. For, resuming the value of/, we may write 24 4 ' 4 Also, when y' is + i o^* — i> /= <^' • Therefore, since 4 (y^ — i)' cannot become negative, wx see that the solution composed of tacks will give the least possible value of t. We have, of course, assumed that it is not necessary to tack back- ward ; that is, that x may always increase algebraically. 228. We naturally inquire whence arises the discontinuity in this class of problems, and why it presents itself in certain forms of /, and not in others. Now the only condition im- posed besides the fundamental one, that the given line shall possess a certain maximum or minimum property, is that it shall also join two fixed points, and if the required maximum or minimum property be not altogether impossible, the dis- continuity must result from imposing the second condition. That it does in general thus arise will appear from the fol- lowing example, in which this condition is removed. Problem XXXVIII. 229. A vessel starting from a fixed point is required to sail a certain number of miles, her speed being ahvays dependent solely upon the direction of her course and certain constant quantities. It is required to determine along what path the given distance may be accomplished in a minimum time. Regarding the ocean as a plane, assume the meridian through the starting-point as the axis of -3{;i,. and employ y^'' ^ — * J-. >.->■ 292 CALCULUS OF VARIATIONS. t and V as before. Then it is plain that we shall have, as for- 4/1 _u/-^ merly, v — F{y') = F, and dt = ^^^— - dx = /{/) dx = fdx. r Hence we are to minimize the expression / fdx, while nx-i, / \/i-\-y'''dx is to remain constant. Therefore the prob- fJ Xq lem is now one of relative maxima and minima, and we have u = fC{f+ ^ ^^+yi ^-- =£? vdx, (I) where it must be observed that V is also a function of y' and constants only. In the- present case, moreover, we do not suppose the second extremity of the required curve to be in any manner restricted, so that x^ and y^ are both variable. Therefore, to the first order, we have SU^ (/+ a ^7 + /') dx, + j/' + -^7=^ I Sy, Whence where rij/'+vf$7-i^-""- « -^ df Now, for the same reason as given in the preceding prob- lem, c cannot, even should discontinuity occur, and the inte- gral be separated into parts, have two values, c^ and ^3, within the range of integration ; and since we know from (3) that c, or the coefficient of Sy^, must vanish, (3) becomes DISCONTINUOUS SOLUTIONS. 293 Moreover, in this case, no relation exists between dx^ and 6>^, because the extremity of the required curve is not con- fined to any other curve, but is wholly unrestricted. There- fore (i) must also give (/+«vr+7^X = o. (5) From (4) and (5) we have / \ and ,^-r^;+y'. (6) vi+yM, y From (4) and (6) we obtain Now since F is a function of y only, we know that the required path must be some right line, or combination of right Imes, so that y is the tangent of the inclination of this line, or else of the last tack, to the axis of x. But it is evi- dent that if the solution can consist of tacks, involving two or more values of y , the arrangement of these tacks will be arbi- trary, since the integral taken through an}' given portion of X will be the same for any one of the tacks — that is, for any one of the admissible values of y — and therefore // can have any one of these values, but no others. Hence, as the pos- sible values of y and 7/ are the same, we may remove the suffix from (7) and write, as the general equation of condition, /'-T^ = o. (8) From (8) we can obtain y in terms of constants only, and it may have one or more real values, the imaginary roots being of course rejected. In the former case there can be but one solution ; but when y has more than one real value, a discon- 294 CALCULUS OF VARLATIONS. tinuous solution by a combination of these values would seem possible. It must, however, be observed that, whether the solution be continuous or not, a must retain the same value throughout U. Now take any two real values of y' found from (8), and make // equal to the first, and 7/, which may be regarded as measuring the slope of some other tack, equal to the second. Then from (6), and also observing that we may interchange the slopes of the tacks at pleasure, we have f Vi+y (9) and as every member in (9) equals — a, we may write / Vi +/■' A-r\ _^ / ^i +/" y' (10) But it will be in general impossible to satisfy (10) by employ- ing any two values of y' found from (8), so that a discontinu- ous solution will not frequently occur. Still such solutions are possible, as we shall prese^ntly show ; and even when no dis- continuity is admissible, it is conceivable that we may have a choice of two continuous solutions, provided j// and // can severally satisfy the equations / / DISCONTINUOUS SOLUTIONS. 2g$ because a in the two solutions need not be identical, but must not change value in the same solution. 230. As a particular example of the preceding problem, let us assume the velocity to be that employed in case 2nd, Prob. XXXVII., so that /and /' will have the values there given. Then by equations (i) and (2), Art. 226, equation (8) of the preceding article will become O. (I) /(/'- --"'+ ± i) — f — .+/■ Therefore, if y be not zero, we have /'-I -1- 2 4 /i — o I 2 • 4 ) or /' + ^'= 4(^1+11. (2) 3 3 ' Whence ^ 3 '1^^121 3 (3) Now if VB be less than -, / will always be imaginary ; if it equal -, / will be zero ; while if it exceed -, one of the values 3^ 3 of y will be negative, and all the real values of y' will be given by the equation y^±V---i.VB. (4) 3 Now we have and CALCULUS OF VARIATIONS. 2 "^ 4 vi +y' Vi +y' fWi+y = {y'-i)Vi+y^ is) But it at once appears that none of the members of (9) will be in any way affected by the successive substitution of two values of y numerically equal but with contrary sign. More- over, in this case, the equation / _rvi+y Vi +y' y reduces to equation (2) ; so that it must be satisfied by either value of y just found, and will also, from what has been shown, be satisfied by substituting the positive value in one member and the equal negative value in the other. Hence it appears that equation (10) of the preceding article will be satisfied by putting for j// and j/ the two values of y given in (4), and by no others. Therefore the solution y = o can only hold when zero is a root of (2), which can only be made true by making d^ = — i, and in this case there will be no other root, and so no discontinuity. But if d^ become greater than — i, we shall have an equal positive and nega- tive value of y, which may be combined in the same solution, thus giving discontinuity. 231. Let us now consider the terms of the second order. These are where '''-i{T)f^'+'''''^'+ir'£^'y''^' (6) 2 4 DISCONTINUOUS SOLUTIONS. 297 Now, because y" is zero, all the successive differential co- efficients of V^ must vanish, and also we have =W-'^^\''- and, as will be seen from equation (4), Art. 229, the coeffi- cient of (^j// likewise vanishes, so that we have left in SU only the terms under the integral sign, and have merely to deter- mine the sign of -—^. Now we have ^= /" + ^ /" = 3/' - I, /Vi+y a ^ y = - (/' - I) Vi +/'. Whence ^^^_3y._i_y^-i_3/*+y dy" ^-^ I +/' I +/' * Therefore it appears that we have a minimum whether y' be positive or negative. When, however, y' is zero, we see from the last equation d'^V that -r-TT is also zero, so that we might infer that this value of dy y gives neither a maximum nor a minimum. But this infer- ence would not in the present case be correct, because we shall find that the terms of the third order reduce also to zero, while those of the fourth order will become positive. It may be also observed, although not affecting the problem, that 298 CALCULUS OF VARIATIONS. when y is zero, dx^ can have but one sign, the negative, if x^ be positive. 232. It will be seen that while the removal of all condi- tions regarding the upper limit does not here destroy the ad- missibility of a discontinuous solution, it nevertheless abol- ishes its necessity. For as the value of /, and also that of x^, will be the same whether we employ the positive or the nega- tive value of /', or some combination of the two, the time fdx will be also unaltered ; and as we are not now obliged to tack in order to go from one fixed point to another, and no time is gained by tacking, the discontinuity is merely admis- sible. The discontinuity in this case appears to arise from the fact that the problem is so constructed that the fundamental equation f -\ „ -^ = c may have two roots, both of which 1/1 +y" give the same value of /, and satisfy all the conditions of the question. 233. Suppose we modify the preceding example byre- quiring that, instead of sailing a certain number of miles, the vessel shall be required to reach a certain degree of latitude in a minimum time. Then we are to minimize absolutely the expression U — J fdx, where / has the same value as be- fore, the limit x^ now being fixed, but jj being subject to varia^ tion. Then we havej as before, f = c, and c cannot have two values. But because ^y^ is not zero,// ot f must vanish, so that we have/^ = y' (y^ — i) = o ; the roots of which arey = o, y'— i,y'— — I. Now asji' is not fixed, we can employ any one of the values of y' alone throughout U. The first will render U a maximum, as we have already seen, while the other two will give ^the same value whether employed sep- arately or in combination, which value is a minimum, as has been shown, and is also the least value of U, DISCONTINUOUS SOLUTIONS. 299 234. We may now consider briefly the inquiry with which we opened Art. 228. Two things affect the problem : first, the particular form of /or V\ and second, the conditions which ai'e to hold at the limits. With regard to the first we may observe that there can be no discontinuity unless /or F be of such a form that the fundamental equation/^ — c can furnish more than one real value of y' . Thus, in Prob. I., the fundamental equation is — -^ = c, which, because s/ \ -^ y is supposed to remain positive, can be satisfied by one value of y' only, so that in this case no discontinuity is possible. Second, when the fundamental equation gives several real values of y, and a combination of them satisfies all the other conditions of the question, the necessity for the employment of this combination, or discontinuous solution, generally arises from the fact that the points to be joined are fixed. More- over, as we in whole or in part remove this restriction from one of the limits, we decrease the probability that these val- ues can be combined at all ; that is, that discontinuity will be possible ; and even when it still occurs, it appears generally rather admissible than necessary. 235. When /is a function oi y" or y^' only, admissible, but not necessary discontinuity is still more likely to occur. Let us consider, as an illustration, a particular case of Prob. IV. Problem XXXIX. Let it be required to maximize or minimize the expression supposing the limiting values of x and y only to be fixed. 300 CALCULUS OF VARIATIONS, Proceeding as usual, we obtain f' = §r = 2[aY-^^=c.^ + c^. (2) But <5>/ and 6y^ are not zero, so that their coefficients // and // must severally vanish ; and assuming the origin at one of the fixed points, we readily see that c^ and c^ also vanish, so that (2) gives «>"- — = (3) and y"=^±-^=±B, (4) Therefore, by integration, we obtain ^=±^ + C,x^C„ (5) in which, because the origin is at one of the fixed points, C^ must vanish, and then C, must be determined by making the parabola pass through the second fixed point, whose co-ordi- nates must satisfy the equation The term of the second order is which is positive for either value of y", thus giving a mini- mum. DISCONTINUOUS SOLUTIONS. 30 1 We have here also the least value of U. For we may write «>"" +yi= [^y - ^7) + ^'^'1'% (7) h h which, by makinpf/^ either + - or > reduces to 2a^U'. •^ a a 236. Here no discontinuous solution can be necessary, because we can always join the two fixed points by a para- bolic arc, in which /' shall be -I — or ; and also, we have a a then the least value of U. Still, a discontinuous solution is always admissible. For Ave can also always pass from the first to the second fixed point by some combination of parabolic arcs, each of which will satisfy (5), but will differ in the values of C^ and C^. Now it is evident that all these arcs, having y" either -)- - or , will satisfy the equation M =0, and it remains a a only to show that they will also make the terms in ^6^ which remain without the integral sign vanish. Consider two of these arcs meeting at the point x^^ y^. The terms arising for this point are But since the equation y = ± - holds for both arcs, f and -^ a dx must vanish for both, thus rendering the expression just given likewise zero ; and similarly for any number of arcs. Here the discontinuous solution consists of parabolic arcs which may meet at finite angles, and the value of U, and also that of the terms of the second order, is the same for either solution. 302 CALCULUS OF VARLATIONS. Problem XL. 237. It is required to determine the solution of Prob. XV, when the length of the given line exceeds that of the semi-circum- ference described upon the line Joiniftg the two fixed points as a diameter. We can of course always, by taking the radius sufficiently great, join two points by a circular arc, whatever the length of that arc may be required to be. But we cannot here ex- tend the arc beyond i8o°; because then there would be be-, yond Jo and y^ both a convex and a concave portion of the arc ; and besides being compelled to count a portion of the area twice, these portions would, as we have seen in Art. 95, give opposite signs to the terms of the second order. Indeed, whatever may be the solution, we would most naturally un- derstand the problem to imply that we are not to go beyond the production of the ordinates y^ and j/, ; that is, beyond the lines whose equations are x — x^ and x = x^, which may there- fore be considered as boundaries which we must not trans- gress. We would therefore feel certain that the solution, at least so far as discoverable by the calculus of variations, can consist only of what will satisfy the equation M —o, with perhaps some portion of these boundaries, unless indeed some other boundary can be discovered. Let us now see what can be obtained in the usual way. We have u = £> + '^ ^' +^1 '''■ =£? f^-^-^' M=,-4^-JL^_, and X ^=_- = .. (I) Now the last equation will be satisfied by y'= co , because we shall then obtain x — a = c, which is therefore a particular or DISCONTINUOUS SOLUTIONS. 303 singular solution, being the equation of a right line perpen- dicular to X. But any such line will reduce M to unity, so that we can only employ one or both bo'undaries joined to a circular arc, because that arc gives the only general solution of the equation M ^= o. Moreover, we cannot assert that c must retain in this case the same value throughout [/, For the terms without the integral sign at either point of junction of the arc and line are of the general form «i(.7r#7)r(-.^-^),H- « which in order to vanish will require that y shall at these points mean the same thing for the arc and the line ; that is, that they shall be tangent. Hence we are not confined to one boundary, but are at liberty to employ both. 238. As the infinite values of y will render our investiga- tions untrustworthy, we must, in order to determine whether the proposed combination be the real solution, transform to polar co-ordinates. Take the pole at any point on the axis of X, between x^ and x^, regarding that axis as the initial line, and denoting by v the angle which any radius vector r makes with this initial. Then it is plain that W must have the general form given in equation (3), Prob. XX 1 1 1., except that the limits will not be the same. For let v^ and z\ be the respective ciUgles which the radii r„ and r^ drawn to the two fixed points make with the initial. Then we need only consider the integ- ral from T'o to -c\, because although all the area in question is not comprised within the limits, still the two remaining tri- angles v/hich are included between the initial and the respec- tive radii and ordinates undergo no variations. We are, then, to maximize the expression ^ =X"' { 7 + '^ ^'-^ + '■'" 1 -^^ =X" ^'^^^ ' (3) 304 CALCULUS OF VARLATIONS. Then, since Sr^ and dr^ vanish, if we suppose U divided as our solution requires, we shall have -\-fJ"M6r dv J^fJ'MSr dv ^-£'M dr dv, (4) where i\/r \ ^^ d ar . ^ S/r'^ + r" dv i/r' + r'' ^^^ Then to make the terms without the integral sign vanish, we must have r/— r/ and r/= r/, which agrees with the result from equation (2). We also know that the circular arc will, so far as it extends, reduce M to zero, so that the second in- tegral in (4) will vanish, leaving only the rectilinear portions to be examined. Now along either of these lines r cos v is constant, so that by differentiation we find , r sin 7^ r = = r tan v, cosz/ |/^2 _|_ ^/2 _ ^ |/j _j_ ^^j^a V = r sec V zo^v r . a r cos V, = sm ^, — , = cos V. Therefore along either of the rectihnear portions M reduces to r. But for these boundaries dr is always negative, so that dC/ becomes a negative quantity of the first order. DISCONTINUOUS SOLUTIONS. 305 Hence, if we vary the whole line, we are sure of a maxi- mum without examining the terms of the second order ; but if we vary the arc only, such examination would be necessary. In this case we can again employ plane co-ordinates, and we have already shown that (^t/ would then become a small nega- tive quantity of the second order. 239. If jKi and jKo be not equal, the arc in a continuous so- lution cannot equal the semi-circumference having as its diameter the line joining the fixed points. Let A and B be the fixed points, and let y^, the ordinate of A, be less than jj, the ordinate of B. Let A Che drawn parallel to x, C being upon the ordinate jj, and bisect AB at D by the perpendicular DE, E being on A C. Then the limit of the continuous solution will be reached when the arc becomes tangent to the ordinate y^ ; that is, when its tangent at A is perpendicular to AC. Then it is evident that the centre of the circle will be at E. Now s being the length of the arc, and R its radius, we shall have the following equations : AD=\^{x,-x:f-\-^y,-y:)\ R = AD sec EAD =:ABVi+ tan= EAD, t^nEAD=^^i^^^-^, •^1 — ^0 Then s can be determined by equation (10), Art. 91. De- note this particular value of s by /. Then if /, the length of the given line, be somewhat greater than /, the line must be first extended along the ordinate y^, produced a certain dis- tance l\ until a point is reached at which the same construc- tion can be made as at ^. Then all the equations just given will be rendered true by merely substituting for y^, y^ -{- 1', so 3C6 CALCULUS OF VARIATLONS. that the new values of R and s may be found in terms of x^, jKo, -^1, y^ and I' , and then we have the additional equation l—l'^s, so that, / being given, V can be also determined. This construction will hold until ^=/i->'o+-(-^i-^o), when the arc will become a semi-circumference. If then / be still further increased, we must retain the same semi-circum- ference, but also produce y^ as well as y^ a certain distance I' , Then we shall have Hence, as / is supposed to be given, I' will be determined, and this construction will hold when / is indefinitely ex- tended. We must, in closing, call attention to the fact that this problem, when discussed by plane co-ordinates as at the be- ginning, affords another instance to show that necessary dis- continuous solutions are generally suggested by the funda- mental equation, even when they do not satisfy at all the equation M —o. Problem XLI. 24-0. It is required to determine the discontinuous solution i?t Prob. XIX. It will be remembered that when x\ is zero, x^ becomes a definite function of the given volume, so that if we require the second point on the axis of x to be fixed — that is, x^ to have a given value — then, unless that value happen to satisfy the equation x^ — V -—-, where tj is the volume, we must resort to some discontinuous solution, if any solution be pos- DISCONTINUOUS SOLUTIONS. 307 sible. (See equation (8), Prob. XIX., observing that c there was shown to equal x^^ Now as ^^in this problem does not admit of the usual transformation, because it contains no variation but that of j/, the fundamental equation is found by equating to zero the co- efficient of 6y dx in equation (2) of that problem, which gives either J/ = o, or else equation (3). This suggests that if the value of x^ be too great — that is, greater than j/ -— — the solution will consist of a curve satisfy- ing equation (4), and extending from the origin to some point x^ on the axis of ;r, x^ being less than ,r,, and then of the axis itself from x^ to x^ ; and that if x^ be too small, the solution may consist of the same solid extended to x„ beyond x^, and then of the axis from x^ to x^, the solutions thus being similar to those in the case of the sphere. Now the terms of the second order, as we see from equa- tion (2), are dU= r^ \ a-\-x-^^^-^'- 1 dfdx. But if we put 7 = 0, and for a its value -^-» we shall obtain 6U^.S \-^-\-—}i^fdx, { 2C^ 2X^ ) where the integral extends over the rectilinear portion only; while if we vary the generating curve, (5^6^ will take the form given in equation (11), where the integral will extend from x^ to x^, and will be negative whether x„ be less or greater than x^. Hence, observing that c — x^, the entire variation may be written 308 CALCULUS OF VARIATIONS. Now in order that U may be a maximum, the second integral in (i) must also become negative, otherwise the sign of dlJ would become ambiguous. But any element of this integral will evidently become negative or positive according as x^ is less or greater than x. Now when the solid does not extend to the second fixed point, x for the rectilinear part is greater than x^, and the same will be true when the solid extends be- yond the second fixed point, provided we agree, as explained in Art. 195, to regard x for the rectilinear part as still increas- ing from x^ to x^ ; so that under this supposition we have always a maximum. 241. But the solution in the case in which the solid ex- tends beyond the second fixed point may not, perhaps, be deemed altogether satisfactory. For in the volume which is generated by the derived curve, we are obliged, as before, in the case of the sphere, to reckon twice that generated by ^y along the rectilinear part, and also to regard its attractive force, when counted the second time, as what it would be if each element were placed as far beyond x^ as it now falls short of that point. We do not, therefore, in reality, compare the attraction ex- erted by the primitive solid with that which would really be exerted by the derived solid, but merely with what the attrac- tion of that solid would be if the attraction of any particle could vary inversely as the square of the estimated value of X, instead of its actual value. Thus we have here merely a sort of theoretical or imagi- nary solution, not properly capable of geometrical representa- tion, and presenting itself possibly somewhat as do imaginary roots in the theor}^ of ordinary equations. But the condition that the solid is to meet the axis of ;ir at a second fixed point may, as Prof. Todhunter has suggested, be more naturally understood to mean that the solid is not to stretch beyond the line whose equation is ^ = x^. Then c in (4) would no longer DISCONTINUOUS SOLUTIONS 3O9 be equal tO-Tj, but could be determined from equation (7) by making the limits o and x^, since x^ and v are both given ; and then all the conditions of the question could be fulfilled. But should neither of these solutions prove satisfactory, we are still at liberty to suppose that there is no solution, since it is evidently possible to assume such conditions in any problem as will render any solution either continuous or dis- continuous impossible ; as, for example, if in Prob. XV. we should assume the given line to be shorter than the right line joining the two fixed points. Problem XLII. 242, It is required to discover the nature of the discontinuous solution in Prob. XXI Here, as will appear from reference to the problem, the continuous solution consists of an oblate spheroid whose major axis is to the minor as ^2 is to i ; that is, whose eccentricity is — =r, ^^ the square of the semi-minor axis, being equal to x^. V2 Hence, if the given volume be greater or less than — , the solution, if any exist, must be discontinuous. But the fundamental equation in this case, as will be seen from equation (4), is y {/ -f 2.r' — 2d') = o, which gives either jj/ = o or equation (5), which is the equa- tion of the generating ellipse. Let A and B be the two fixed 3IO CALCULUS OF VARIATIONS. points on the axis of ;r, and C the origin, which, it will be remembered, was required to be midway between A and B. Then it is suggested that the discontinuous solution might be that represented in the figure, where the generating ellipse is DE or FG, according as the given volume is less or greater than -. 3 Here, then, U for either case may be divided into three in- tegrals, extending respectively from x^ to x^, from x^ to x^, and from x^ to x^ ; x^ being in the first case the abscissa of D, and in the second that of i% and x^ being that of E or G. We must also recollect that in the second case x^ and x^ are thus estimated : X, = - (CF+ FA) and x, = CG+ GB. Now we have seen (Art: 120) that the terms of the second order for the ellipse reduce to / ysydx, and to obtain the variation of the rectilinear portions we have merely to make f zero in the first equation of that article, so that we have d[/ =fj\^" - a')6/dx + fJ'/S/dx + fj\x' - a'')6/dx. To render the first and third integrals positive, we must have x"" > d^; and since d" = x^ = x^, if we estimate x for the recti- linear part as already explained, this condition will be fulfilled in either case, and ^becomes a minimum. But since the solids generated by both the primitive and the derived curve are to be revolved about the axis of j/, it must appear that when the sohd extends beyond A and B the solution, like that of the preceding problem, is merely theo- retical or imaginary. These problems also resemble each other, and differ from all others which we have considered, in that, as t/ contains x and 7 only, there are no terms m 6U DISCONTINUOUS SOLUTIONS. 3II without the integral sign ; and hence the equation L = gives, without integration, the equation of the required curve, and there are no terms to consider at the limits. Problem XLIIL 243. // is required to determifte what discontinuous solutions can present themselves in the discussion of Prob. XX. Here the continuous solution is an hypocycloid, in which the radius of the rolling circle is one third that of the fixed circle. But, by the closing remarks of Art. 1 16, it appears that this solution cannot hold when the given volume becomes less than ^, where b is the radius of the given base ; so that if the given volume be less than this quantity, the solution, if there be any, must be discontinuous. Let AD be the axis of x, and DB the radius of the gen- erating base. Then, since the volume was to be upon the given base, we would naturally infer that when the volume becomes too small, the generating curve would consist of an arc AC oi an hypocycloid, and a portion CB of the radius of the base. In fact, we may understand the conditions of the problem to imply that the solid is always to be upon a portion of the base. 244. Before considering whether this solution is also sug- gested by the calculus of variations, we will show that it is in some cases the solution required. 312 CALCULUS OF VARLATLONS. For the solid generated by ^(7 the resistance will evidently be 27tj^ "Y /2 ^^^ ^^<^ fo^" the ring generated by CB it will be 7t{b'^ ~ yi)' Hence we may minimize the expression where we are to regard jj, the ordinate of C, as variable, but the other terms at the limits as fixed. Now taking the varia- tion of U under this supposition, transforming it as usual, and making M vanish, we shall obtain, as in Prob. XX., equation (4), which will be of course the differential equation of the hypocycloidal arc AC. But we have, after malting J/ vanish, and to satisfy this equation, we must, since y^ is not zero, have + (I -^ry'J °' which gives y = ± i' Thus it appears that the generating curve must meet the ordinate of B at an angle of 45"". 246. To determine the sign of the terms of the second order, we must observe that the terms under the integral sign in the value of ^ given in (i) equal 2^ in Prob. XX. Hence we shall obtain from the variation of these terms twice the second member of equation (19), Art. 117. But we shall also obtain from this integral a term without the sign of integra- tion. For (19) was obtained under the supposition that 6j/, and (^Ko vanish. When, however, this is not the case, we must, as we see from equation (6), Prob. VHI., add to the second member of (19) the terms J (//<5j," -•/.%."). (2) DISCONTINUOUS SOLUTIONS. 3I3 which will give in this case the additional term HI +/ ) ' 1 and as this is cancelled by the term of the second order arising from the variation of — y^ in U, SU becomes merely twice the second member of (19), which is positive. 246. Thus we have a minimum if c have any real value. Now because j/ = ± i, taking the positive sign, we have, from equation (4), Prob. XX., which, it will be remembered, is the c fundamental equation in this case also, jj/j — — ; and it is also 4 shown by operations of the differential and integral calculus only, that the given volume, v', will in this case be v' = -^ (3) 1920 ^^'' Hence, when v^ is given, c and j/j are at once determined. Now v' can be given as small as we please, but it cannot be as great as we please. For j\ must not exceed d ; and as c = 4jj, v^ evidently increases as we increase y^, and must have its greatest value when ji = b — that is, when c :=4d — and then (3) gives .-135^. (4) 30 ^^^ We see, then, that if the given volume be less than -, we must always employ the discontinuous solution ; if it be s^reater than — — , we must alwavs employ the continuous 30 . ^ J solution ; but if it He between these values, then we shall have two mmima, and must determine which will give the smaller 314 CALCULUS OF VARIATIONS. resistance. This determination must, however, as in former cases, be effected by the ordinary calculus alone, using, of course, any equation which has been thus far obtained. It will be sufficient here to give the necessary formulse and results. Let 2\ denote the angle whose tangent is jk/. Then, R being the resistance. Prof. Todhunter shows, by methods of the ordinary calculus, that for the continuous solution „ __ nb^ (i sin^ v. ^-^^i. (5) cos v^ \ 4 and that for the discontinuous solution where, since v' is supposed to be a given quantity, v^ can be determined from equation (12), Art. 116, and c from equation (3) of this article. Now if we take the extreme values of v', for which two solutions are possible, ^' — ± and v' = -^ , (7) 5 30 ^^^ we shall find that the two solutions coincide for the first, R being in either case ^- — -, and for the second value of v' we 20 shall find that R is less for the discontinuous than for the con- tinuous solution. For we have in the first case I _ ^ — £_ K — Ttb^ X. 440 1 2, nearly ; and in the second _ 9^^' 20 DISCON-TINUOUS SOLUTIONS. 31 5 It is also shown, by determining the sign of -— , that both for dv the continuous and discontinuous solution R decreases as v' increases. Hence, from what has been already shown, it will appear that, when there are two solutions, the discontinuous is that which will always give the smaller resistance. 247. It will be remembered that in Prob. XX. we con- sidered only the case in which v is supposed to be zero when y is zero. But if we supposed that when y is zero v is — , and measure the arc s from that point, then we shall have, from equation (lo) of that problem, ^ = cos yu. Here, on account of the infinite value of y\ our investiga- tion of the terms of the second order will not be satisfactory, and we will therefore adopt y as the independent variable. Then C^ becomes Hence, to the second order, Therefore, by making the terms of the first order vanish, 2yx' 2ay'^ — - — Y^ — — = a constant, which must be o ; (I -j- ;r ) and this must, of course, lead to the hypocycloid, as before. Then, as the terms of the first order vanish, we shall have '^^=r>(S^<^-'^^-^' 3l6 CALCULUS OF VARLATLONS. which is negative so long as x'"^ does not exceed - ; that is, so 7t long as ^1 is not less than - Thus in this case the resistance becomes a maximum, provided we can determine real values for c. Now, as before, it is shown that in this case cos^ v^ 1 3 cos^ 2^ 1 5 cos* v^ 7 cos'^ ^1 _|_ i ~3 10 ' 8 6 ^3 , . V = 7tc COS V, -^ I , -^ . ( 10) Also, because equation (5), Prob. XX., holds, we shall find that here, as in equation (13) of the same problem, ^ = sin' z/j cos 2/j, (11) and from these two equations c must be determined. It is evident that v' can be made as small as we please ; but it can- not be taken as great as we please, because it decreases with v^ ; and in order to have a maximum, v^ must not be less than (12) which is therefore the greatest admissible value of v' . 248. We are naturally led to inquire whether there will be any discontinuous solution when v' exceeds the value just given. Since the solid is to be bounded by the given base, the only suggestion which presents itself is that y^ may now be greater than b. Then, when y is the independent variable, U will have the form given in (8). But now, as 7, is variable, we must, when we vary U, increase also the limit y^ by dy^ ; 7t 3* But when v^ = -, we shall find , 32171^ I215 DISCONTINUOUS SOLUTIONS. 317 that is, we must add to the terms of the first order in (9) the term V, dy, or | YjIV' + ^^^'^' \ ^^'' Now the coefficient of Sx^ will necessarily vanish, but we can- not also make V^ vanish. Hence SU \.o the first order will not vanish ; and as dy^ may have either sign, U will be neither a maximum nor a minimum. 249. A somewhat curious point is here noticed by Prof. Todhunter, which it may be useful to consider. Let A be the distance of the base from the origin. Then we may evidently consider the solid as composed of cylindri- cal shells whose radius is y, thickness dy^ and length A — x. Then, instead of £yfdx or fjy/^'dy, the volume may be written / ^27ry{A — x)dy. Therefore with this value of v' we are to maximize or minimize the expres- sion +J2'\-"+'^jj^f\""l>- (.3) Hence, by integration, we obtain — a/-\- ^ — a constant, which must be o. 3l8 CALCULUS OF VARIATIONS. This equation is in reality the same as that which we ob- tained before, and leads, therefore, to the hypocycloid. Thus the integral in (13) will vanish, and so also will the terms at the lower limit, because there y is zero ; but the terms at the upper limit will not vanish, so that we have, by the last equa- tion. Now since the base is a boundary which we may not pass, dx^ is essentially negative, and thus ^U becomes a positive quantity of the first order, indicating that we have a condi- tioned minimum, which result would seem to show that we can never have a solid of maximum resistance, thus conflict- ing with what has been before proved. 250. To explain this difficulty, let AB be the primitive curve, and suppose we wished to pass to a derived boundary EDB, where DB is parallel to x, and infinitesimal. Then we could not derive this boundary from AB by inhni- tesimal changes in y and y, although we could by such changes in x and x'. This assertion, which Prof. Todhunter takes no pains to establish, may at first appear incorrect, because we seem to have given x^ a finite variation in order to obtain DB, which would be inadmissible. But the position appears to be sound, since we should regard x^\ after being varied, not as the tan- gent of the inclination of DB to y, but as that of the inclina- tion to y of the tangent to the derived curve at D, supposing DISCONTINUOUS SOLUTIONS. 319 the curve ED produced beyond D. Then Sx^ need not be finite. Now since y^ is fixed, we shall (unless in the last article we make dx^ zero, in which case all the terms of the first order will vanish, and there will be no difficulty) be obliged to pass to a derived curve terminated by DB, DB being numerically equal to Sx^. Still, so long as we adopt for the volume, as we did in (8), the expression nj y'x'dy, we cannot pass to such a boundary as we have been considering ; because although the expression just given will represent the volume generated by the primitive, still, when we change x' into x' -\- dx\ and write v' = TtJ y{x' -\- dx')dy, v' can only represent the vol- ume generated by ED, neglecting entirely that generated hy DB. Hence we conclude that the form of v' adopted in (8) is not general enough to permit of a full discussion, as it will not allow every change in the form of the solid which the calculus of variations would in this case sanction. We see, also, that we can have a solid of maximum resistance only under the condition that jKi, the radius of the generating base, shall be invariable, and that the curved part of the solid shall always extend to the circumference of the base. 251. We have in this discussion a remarkable confirmation of the principle often before stated — that when by variations we have obtained conditions which render any definite inte- gral U a maximum or a minimum, we are not necessarily war- ranted in asserting more than that C/ is a maximum or a mini- mum with respect to admissible variations. For the sohd of minimum resistance which we obtained in Prob. XX. is not the solid of least resistance, since by taking a zigzag boundary it could be still further diminished, although we could not pass to such a boundary by the calculus of v? nations. More- over, our solid of maximum resistance is such so long only as 320 CALCULUS OF VARIATIONS. we do not make suck a change in the form of the sohd as in Art. 240. But a solid of still greater resistance would evi- dently be obtained by passing to a boundary in which y' is alternately zero and infinity, although such a change of form cannot be effected by the calculus of variations. 252. It will be remembered that in Art. 243 we were led to the discontinuous solution, which we subsequently verified, by the consideration that the given base constituted a bound- ary, and that therefore it would probably form some portion of the solution. Now we have found hitherto the boundaries to be also in some manner suggested by the fundamental equation which IS usually the first integral of the equation M — o, even when these boundaries do not in reality cause M to vanish at all. In the present case, however, the discontinuous solution does not appear to be very clearly suggested by the calculus of variations alone, unless, indeed, U can be put under some form different from those which we have yet examined. For, adopting in succession x and y as the independent variable, the first integral of the equation J/ = o will be in each case the most general form of. the fundamental equation, and we shall have yy ^y — 7 I /2^2 = a constant (i -\-y'J and y^' ^^ ~ { i-\-x''f "^ ^ constant. which constant must, in either case, be zero, because the curve is to meet the axis of x. Therefore, rejecting the solution J/ — o, we have = /' ^ //' H - ""' - ^'^' and these equations lead to the same solution. DISCONTINUOUS SOLUTIONS. 32 1 Now y = CO or x' = o are not solutions of these equations, unless, indeed, we could suppose <; == co and C — co . But these constants will not be infinite for the curve ; and since they are in each case the reciprocal of a, if we remember that even in a discontinuous solution the constant introduced by Euler's method cannot, like a constant of integration, have two values, it will appear that c and C cannot become infinite at all. 253. There would seem to be nothing surprising in the fact that the fundamental equation does not always suggest a boundary which does not cause M to vanish at all, and indeed it would appear more remarkable that such boundaries are so frequently suggested. Cases, however, like the present ap- pear to be rare, and we have now had abundant proof that the calculus of variations does usually suggest solutions when they are possible, and even when such suggestions would not naturally be expected. Moreover, in discontinuous solutions it very often happens that a trial solution is easily reached without the aid of varia- tions, or at least without examining the form of M; and then the calculus of variations affords us the means of verifying or falsifying this proposed solution, and that, too, very frequently without an appeal to the terms of the second order. 254-. The subject of the present section has been most elaborately treated in the Adams Essay, or Researches in the Calculus of Variations, published by Prof. Todhunter in 1871, and to his labors its present degree of perfection is chiefiy due. In this section, which is little more than a condensed view of that treatise, we have endeavored to present all the leadmg points of that work, and particularly those points which were new to our science. All the examples, there- fore, of this section have, with slight modifications, been taken from this essay, although we have in no respect followed its 322 CALCULUS OF VARLATIONS. order. We therefore earnestly recommend the work to all who wish to become fully acquainted with this subject. We have, with the exception of Prob. XXXIII. , consid- ered those cases only in which the discontinuity ma}^ be sup- posed to arise from conditions unconsciously imposed, or at least imposed without seeking to produce it ; because it is only when it thus presents itself that its origin can be a source of difficulty. It must, however, be evident that even when no discontinuity would naturally occur in a problem, we can easily impose such restrictions as will render a discontinuous solution necessary, and the work to which we have referred is occupied largel}^ with such examples, some of which exhibit much ingenuity. But as these examples, while affording ex- cellent practice in this department of analysis, present nothing which has not been already considered, it will be sufficient in closing to subjoin one, which is all that our space will permit. Problem XLIV. 255. It is required to find the path of quickest descent of a particle from a fixed point A to a second fixed point B^ under the condition that the path is not to pass without a given circular arc AB, which is not to exceed a quadrant ; the particle starting from a state of rest at A , and B beijtg the lowest point of the arc. Assume the horizontal as the axis of x. Then, as in Case 2, Prob. II., we shall have ^^0 Af7. t/.Tn ' 6U=:P6y-\-fMdydx, (I) DISCONTINUOUS SOLUTIONS. 323 where the limits and suffixes are for the present omitted, and p^dV dy ^yix^y"^ (2) ax 2j'i ax where iV^== -— . Now wherever the sig^n of ^y is unrestricted, ay J/ must vanish, and this will lead to a cycloid having- its cusps on the horizontal line through A, and its vertex downward. But the cycloid alone can never be the solution, because its tangent at A being perpendicular to x, it is initially with- out the circle. Since, then, the circle is the only boundary along which the sign of Sy can be fettered, the solution must consist either of the given circular arc alone, or of, first, a portion of that arc beginning at A, follow^ed by some combina- tion of portions of that arc and cycloidal arcs given by the equation J/ = o. 256. Let the initial and the first cycloidal arc meet at the point x^, y^. Then there will evidently arise in dJJ., as given in (i), the terms [P^ — P^Sy^, 2iXidi this must either become positive or vanish ; that is, since Sy^ must be negative, P^ — P^ must be negative or vanish. But if it were negative, we would, as appears from (2), have y^' >y^', which would re- quire the cycloid at that point to pass without the circle, which is inadmissible. Hence the coefficient of Sy^ being zero, we have yj =^yjj and the circle and cycloid must be tangent at the point x^, y^. In like manner they would evi- dently be tangent if they could meet at any other point. 257. Let AC be the horizontal through A, O the centre 324 CALCULUS OF VARIATIONS. of the given circular arc, and r its radius, R being the point x^yv ^o that RTisdi normal to the cycloid. Now take any point on ^(7 as the origin. Then the equa- tion of the circular arc is {x-cr+{y + bY = ,^, (3) where b = OC, and c is the abscissa of O. Therefore, for the circle, we find f=-- X — c y+b' Hence, from (2), we have N Vi +/' = y-^b (4) ~ r X — c 2j/^{j/ -\- b) 27y\y -f- b) r Vy dP_ -2y+y(x-c ) _ -2y(y + b)-{x-cy dx ~ 2ry^ ~~ 2ry^{y + b) Whence, putting in N the value of r' from (3), we have y — b (5) J/: 2ry^ (6) Now this value of AI must either vanish or become negative in order that / MSydx may be positive along the circle, since DISCONTINUOUS SOLUTIONS. 325 M will vanish along the cycloid, and this requires merely that y — b shall not become positive. 258, Let / and a denote respectively the angles OTC and OAC. Then RT=OR-OT=r--J- = r- ^^^^- ; (7) sm t sm ^ ^' ^ and because i^T'is a normal to the cycloid, we have, D being the diameter of the generating circle, ^ RT sin t — sin a , ^ D = -^- = r ^^ . (8) sm/ sm / ^ ^ If the cycloid can meet the circle again at some other point S, we shall obtain a similar expression for D, only t will then denote the angle which 6^5 would make with ^(7, and these expressions would be equal. Hence, regarding / as variable, and writing: v = ^- , we must be able to ^ sm'/ effect that v shall twice have an assigned value, or else the circle and the cycloid cannot meet more than once. Now we find dv 2 sin a — sin / , ^ — = cos / r-3- . (10) That is, to render v a maximum or a minimum we must have either 2 sin RT, Whence OT sin TOE = EF = b is greater than RT sin RTF or RF sin TOE ; that is, ^ > y and y — b is negative. 260, We must now show that a cycloid can be drawn tangent to the circle internally at R and passing through B. DISCONTINUOUS SOLUTIONS. 32/ First, assume D — CB z= r — b, putting the vertex at B. Then, since 2(r — b), the radius of curvature at B, is greater than r, the cycloid will be tangent to the circle externally at B. But by taking D sufficiently large, the cycloid still passing through B, we can cause the cycloid to fall entirely within the circle, and then by diminishing D, while retaining ^ as a cycloidal point, we must arrive at a value of D which will cause the cycloid to become tangent to the circle before cutting it, and this point of contact will be neither at A nor B. For at A, y' for the cycloid is infinite, while for the circle it is not ; and at B^ y' for the circle is zero, while for the cycloid it is not. Now as the solution is real, it is unnecessary to discuss the value of D or the position of the point of contact R, or of the cusps on A C. 261. No natural discontinuity presents itself in the discus- sion of Prob. II., since, if the two fixed points be not in the vertical nor in the horizontal line, we can cause a cycloid to pass through them both, and have its cusps on the horizontal line through the upper point. Neither can there be admissi- ble but unnecessary discontinuity of the kind discussed in Prob. XXXIX. For if there could be two cycloidal arcs meeting at any point, they must, as we have seen, both have their cusps on the horizontal through the point from which the particle starts, and must also, as appears from Art. 256, be tangent. Moreover, from Art. 25, the fundamental equa- tion is j(i +y^) = ^ = i^; and since y' has but one value at the point of contact, D can have but one value there for the two cycloids, and the cycloidal arcs must therefore be gen- erated by the same circle rolling on the same horizontal. Now as y' in any cycloid can have a given value but once, these arcs have also their cusps in common ; that is, there are not two cycloidal arcs at all. 328 CALCULUS OF VARIATLONS. Section X. OTHER METHODS OF VARIATLONS. 262. Hitherto, whether employing plane or polar co- ordinates, we have ascribed variations to the dependent vari- able only and its differential coefficients, adding also, when a change in the independent variable is necessary, an increment to its limiting values only. This method, which has been adopted by the two most elaborate English writers. Profs. Jellett and Todhunter, as also by the chief German writer, Strauch, is undoubtedly the best. But many writers vary the independent variable also throughout the whole definite integral ; and as the reader will be likely to meet with this method, the present work would be incomplete if it did not explain this method sufficiently to enable him to follow the solution of any problem in which it may be employed. First Method. 263. Suppose we assume the equation U^ r^Vdx, ■ (i) where Fis any function of ;r, j, /, etc., and suppose j;/ to be- come the ordinate of some primitive curve. Then, by varying ^in the most general manner, we can pass to any curve which can be derived from the first by infinitesimal changes in .t'o, x^y y, y, etc. But we may also pass to the same derived curve by mov- ing, without change of value, any ordinate of the primitive curve an infinitesimal distance djtr along the axis of .v, and then varying it so as to make it become the ordinate of the derived curve for the new abscissa x + ^-^. In this method (^j/, ^y, etc., will mean the difference between/,/', etc., for the primi- OTHER METHODS OF VARIATIONS. 329 tive curve, and corresponding to the abscissa x, and the same quantities for the derived curve corresponding- to the abscissa X -\- ^x. Of course for any given value of x we may suppose f^x\.o have either sign, or to vanish ; and it is evident that when the hmits are to be fixed, the latter supposition must be made regarding the quantities ^x^ and Sx^, 264, We are led, then, to inquire what will be the expres- sion for dU, when x also is regarded as capable of variation throughout the definite integral U. In (i) change x into x-{-dx, y into y -\- Sy, etc., and let U' = [/+ 6U and V = V+ dF (2) be the new values of [/ and V. Then, observing that dx will become, by being varied, Sd.v = -^_(.v + ^-r)dx, (3) we shall have ^' = X''^"^^"+<'-)^--- (4) Whence ^^^ dx ^^^ =jiy ^^+ ' ^) i (- + <^-) '^'^ -£' ^'^-- (5) This is exact ; but approximating to the first order only, we have 330 CALCULUS OF VARIATIONS. where brackets denote the complete differential coefficient of F; that is, Moreover, it is evident that, to the first order, 6V^ Mdx + Ndy-^Pdy' + etc. (9) Hence (6) becomes du^ K6x, - VM.+X^' ) my+p^y+ Qdy^+ etc. - {JVy + Py" + Qy'" + etc.) dx \ dx, (10) 266. But the formulae hitherto emplo3^ed for Sy\ dy'\ etc., will not now hold true, so that we must, before we can fur- ther transform (10), ascertain what will be the values of these quantities under the present supposition. First, in y change x into x-\-dx and y into y -\- dy^ and we have sy-^Ay±M-yJ ^/ -y . "^ dx dSy y'ddx . . dx dx , _ fdSy y'dSx\ f , dSx\ — ^ , v ^ ~^ dSx ^~\dx dx IV ' dxj dx which is exact ; and to approximate to any order required we have only to develop sufficiently the second factor. Thus, to the second order, OTHER METHODS OF VARIATIONS. 331 or, omitting the terms of the second order. To obtain to the first order the value of ^y" we have only to substitute in (13)/ ior y,y" ior y', and y ior /\ so that 6y"^^{Sy-y"Sx)+y"'6x = £, {^Sy - /Sx)+ /"Sx. (14) The Greek letter &? (omega, or o) is usually put for 6y — y'6x. Then we shall find .y = g +/'.., sy' = '-^+y",., (15) which equations are, of course, true to the first order only. 266. Now substituting in (lo) the values of ^/y^y^ etc., derived from (15), that equation will become dU= V,dx- V.dxo +Xy^'^ + ^^' + Q^" + etc.)^;tr, (16) where od' = — -, etc. Here oo, od' , od", etc., take the place of ax ^ Sy, dy', 6y'\ etc., in the former method, so that integrating by parts, as in that method, we shall obtain dU = V,Sx^ — V,Sx, + k^oo^ — h,0D^ + i,Go^' — i,oo^ + etc. +£?^^- P' + Q" - etc>^^, (17) where the coefficients of, (y^, od^, go/, etc., are the same as those of ^y„ Sy,, Sy/, etc., in equation (5), Art. 36, /i, t, etc., being 332 CALCULUS OF VARIATIONS. used as in equation (7), Art. 37 ; while the coefficients of Godx and Sydx are also identical. Moreover, since dx^, dx^ and ^x^, dx^ mean the same thing in the two methods, it appears that (^'^in this case is the same in form as the most general variation of U found by the other method, go taking the place of fy. 267. Suppose, now, we wish to discover by this method the conditions which will maximize or minimize U. Then it will appear, by the same reasoning as before, that (^C/to the first order must vanish, while the terms of the second order must preserve an invariable sign, becoming negative for a maximum and positive for a minimum. Hence (17) may be written SU=L,-L,+X^'Mcodx = L,-L, ^£^'Mdydx —fJ'MySxdx = o. (18) Therefore the coefficients of Sydx and dxdx are so related that if one vanish the other must vanish also, unless, indeed, y should become zero throughout the curve. Now ^x and fy under the integral sign are entirely inde- pendent of any conditions which those quantities may be re- quired to fulfil at the limits, and therefore we must have L^ — L^ = o and / Moodx = o. (19) But QD, like Sy, is wholly in our power, Avhile M \s not, so that we must necessarily, as before, suppose M to vanish, and we can obtain no additional equation by considering separately the integrals in the last member of (18). 268. Let us now briefly consider the terms at the limits. Suppose, in the first place, x^^ x^j y^y /o> • • • • Jo^**""'^ to be OTHER METHODS OF VARIATIONS. 333 fixed ; that is, to have no variation. Then qd^, gj^, go/, etc., and Sjt:^ and ^x^, will severally vanish. For let y"*) be an}' dif- ferential coefficient of j not higher than y** - 1). Then we have at either limit and dx being zero at either limit, we have for that limit d^QD ,. — Gj(^) = O. dx Hence, in this case, L^ — L^ will vanish, and we must deter- mine the 271 constants as we did formerly when all the limit- ing values were fixed. Let us next suppose x^ and x^ only to be fixed. Then, at either limit, od = (^J/, od'= ^y, oo"^ dy" , etc., and assuming these quantities to be unrestricted, h^, h^, i^, /„, etc., must severally vanish, which are the same conditions for the determination of the constants as we would have under the same supposi- tion by employing the other method. Neither can we ob- tain any additional equations by putting for cj, gd\ etc., their values, and then making Sx at the limits vanish. If we make the limiting values of y also invariable, c^^ and gd^ will vanish, all the other conditions remaining as before, so that we shall only lose the equations //, == o and h^ = o, which will be re- placed by the conditions that y^ and y^ must have given values. Proceeding similarly, it will appear that when x^ and x^ are fixed, the same equations for the determination of the 271 arbitrary constants arising from the integration of the equa- tion M = o will be obtained as would, under the same sup- position, have been found by the other method. Let us, in the last place, suppose that x^ and x^ are also variable. Then, if no restriction be imposed upon any of the variations, we shall have, besides the equations already ob- tained, V^ = o and V^ = o, and we shall find that we cannot 334 CALCULUS OF VARLATLONS. obtain any other equations. Here the conditions are the same as those noticed in Art. JJ, and the 211 -f 2 equations cannot in general be satisfied. But suppose that, as in Prob. IX., the extremities of the required curve are to be confined to two fixed curves whose equations are, as in Art. 69, jj/=/and y — F, f and i^ being functions of x. Here Sy has not the same meaning as in the former method, so that equations (10), Art. 69, or rather equa- tions (2), Art. yG, will not now be applicable. But it is evi- dent that now Sy^ ^z^f^dx^ and dy^ = FJSx^; so that we shall now have at the upper limit ^r = (// - j/) ^^^, ^/ = ] #- ir - /) \ ^^. (20) dx^-" -^ M and similar equations in F hold for the lower limit. Now observing that dx^ and dx^ here mean the same thing as dx^ and dx^ in the other method when used to change the limiting values of x, we see from equations (2), Art. 76, that for either limit we must substitute the same thing for Sy in the first method as for gd in the second, and the same thing for dy' in the first as for gd' in the second ; so that the coefficients in- volved must evidently be the same in both methods. Hence we must always obtain by either method precisely the same equations of condition at the limits. 269. Thus it will be seen that the results obtained by the two methods are the same, whether as regards the general solution, or the conditions which must hold at the limits, and that nothing is gained by the latter method, while the labor of obtaining the required results is somewhat increased. This disadvantage will become still more obvious when we seek to examine the sign of the terms of the second order. We shall not, however, enter upon this examination further than to observe that we must, in finding these terms, be care- ful not to reject any of the terms of the second order. Thus, OTHER METHODS OF VARIATIONS, 335 after having approximated to the second order in equation (5), if we employ (13) and (14) in transforming the terms of the first order, we must remember that the value of Sy which we now require is given by (12), and that (13) and (14) are not sufficiently accurate, and that we must therefore add to the terms already assigned to the second order those which are neglected in the first by the use of (13) and (14) ; and it is easy to see that this will generally involve us in much diffi- culty. It is believed that the foregoing account of the present method will be found sufficient to enable the reader to follow any solution which may be presented, which is all that is necessary, since its adoption, as a mode of original investiga- tion, cannot be advised. Second MetJiod. 270, The method which we next proceed to explain pos- sesses oftentimes decided advantages, particularly when we come to consider problems involving three co-ordinate axes, and is moreover that which is adopted by Prof. Jellett in the discussion of geometrical problems. As we shall be obliged to consider it at some length, the reader will, we think, most easily comprehend its nature and use by the consideration of an example. Problem XLV. // is required to discover the co?iditions which will maximize or minimize the expressioji U ^^ j vVi -\- y''^dx, zvhere v is any function of x and y only, and constants , the limits being fixed or variable. Now assuming s as the arc of the required primitive curve, C/'may be written u^ r^vds. (i) 33^ CALCULUS OF VARIATIONS. Let ab be the required arc, and on it take at pleasure any points c, d, e, etc., and regard these points as knots or spots upon a flexible cord. Then, when we make any infinitesimal altera- tion in the form of ab, the arcs ac, ad, ae, etc., will undergo no change in length, but the co-ordinates of the points c, d, e, etc., will in general undergo an infinitesimal change. But the arcs ac, ad, etc., are any values of s, measured from a, so that it appears that we can pass from ab to any derived curve by varying x and j/ in (i), while regarding s, and there- fore ds, as undergoing no variation. 27!. Taking the variation of (i) under this supposition, we have But (2) gives the variation of U only under the supposition that we need not make any change in the length of the primi- tive curve, which is not usually the case. For suppose the required curve be conditioned to always connect two fixed points or two fixed curves. Then if we vary ab without pro- ducing any change in its length, we shall in reality reduce the problem to one of relative maxima and minima, in which the length of s is to be fixed, and in which, as we have already shown, the form of the derived curve cannot be wholly unre- stricted. If, then, the problem be, as we have assumed, one of absolute maxima and minima — that is, if we are required to vary the form of ab in the most general manner consistent with the method of variations — the arc of the derived curve connecting the given points or given curves will not neces- sarily have the same length as ab. Still it is not necessary to vary s or ds under the integral sign, because we can evidently pass from ab to any derived curve AB by first, before varying ab, giving to it increments or decrements at a and b so as to obtain a new arc equal in length to AB, and then varying the form of this new arc in the most general manner. OTHER METHODS OE VARIATIONS. . 337 But as these increments must be infinitesimal, we may de- note them by ds^ and ds^. Now if in (i) we change the hmits into s^ -[" ^^^ ^^<^ -^1 + ^^v we may find approximately the change which will result to U in precisely the same manner as if the expression were U =^ I Vdx, and x^ and x^ only were to be varied. Hence this change will be V, ds, - V, ds, + \ \^^f'^ - \ \j^f'^' + ^^^^ (3) where brackets denote the total differential coefficients. But we wish to find t^^to the first order only, so that we may write, as the new value of U, U' ^^ U -\- z\ds^ — v^ds^ = i\ds^ — v^ds^ -|- / vds. (4) If now we vary the form of the arc in the most general manner, and suppose U' to become U'^, U" will exceed U' by the second member of (2) increased by dv^ds^— Sv^ds^. Hence, observing that the last two terms, being of the second order, must be rejected, wc shall find u"-u=su= V, ds, - ., ds, +X" 1 J s-- +j/y\dS' (5) which is the form of (^"^ which we must in general employ whether the curve be required to connect two fixed points or two fixed curves. 272. As Sx and 5y now denote the changes which the co- ordinates of any point when regarded as fixed on the arc, like a knot on a cord, would undergo, owing to any infinitesimal alteration in the form of the arc, it will, we think, appear after a little reflection that we cannot regard ^x and Sy as entirely independent, although we cannot state explicitly the nature of the relation subsisting between them. We can, however, 338 CALCULUS OF VARIATIONS. easily assure ourselves that they are not independent. For if they were, then, since d 6^ must vanish in order that f/may be y-i-idv jydv a maximum or a mmimum, we would have -- = o and -- = o. ax ay Whence dv dvdy _ Vdv~\ _ dx dy dx VjdxA Therefore we would have as a condition necessary to a maxi- mum or a minimum -^^ = a constant, which is false, since in Prob. VII. we have u ^ ly ^Y^y- dx ^ ly ds, and y is not constant. If we could express ^y explicitly in terms of Sx and other quantities, we might eliminate one of the variations, and then equate the coefficient of the remaining variation to zero. But as this cannot be done in the present case, an ingenious method of overcoming this difficulty has been devised by La- grange, which we now proceed to apply, reserving a general explanation of this method until the reader has become some- what familiar with its spirit. 273. We have always, whether along the primitive curve ab or the derived curve AB, ds" — dx^ -\-dy^^ so that ^-+/^_i=o, (6) where accents will denote differentiations with respect to s ; and as this equation must always hold, it follows that the vari- ation of its first member — that is, the change which that mem- ber will experience when we change x into x -\- ^x and y into y-{- ^y, s remaining unaltered — will be zero. Hence we must have x'dx' -\-y'dy' = o. (7) OTHER METHODS OF VARIATIONS. 339 Then, when we change x into x -\- Sx, the new value of x' is -(.+ / - o. (21) Substituting in the first of these equations for //jk/ its value, — x/, found from the second, and observing equation (6), we obtain, as before, ^j— /^ = o ; so that here also, as appears from (16), v = L If /j should become zero, then, since neither x/ nor y/ can become infinite, the upper hmiting terms would reduce to 344 CALCULUS OF VARIATLONS. v^ds^ = o, SO that v^ must also vanish. Hence, here also, c in (i6) vanishes, and we therefore have always v — L 280. The reader of Prof. Jellett's work will observe that in Chapter IV., in which he adopts this method, he has, in giving the terms at the limits, uniformly omitted the terms V^ds^ — V^ds^y and this omission has led him into an unsatis- factory method of determining the constant c, which is in his book a, and which, as we have seen, can be determined regu- larly by the equations at the limits. (See Todhunter's His- tory of Variations, Art. 348.) It happens, however, that his results in discussing by this method the conditions which must hold at the limiits are in every case correct, although the method by which they are obtained is certainly not strictly so. The reader will find it profitable to verify this latter assertion, which is made upon the authority of the author alone. 281. Let us now return to the general solution. Putting V for /in the last members of (14), we have vx'', Vy — yv' = vy". (22) Now in these equations multiply v^ and Vy by ^''^+y^ which is unity, and put in each for v' its value from the first of equa- tions (15). Then, reducing and factoring, we shall obtain y'i:i'xy' — 'Vyx') = vx", x'(vyx' — v^y') = v/\ (23) Multiplying the first of these equations by y\ the second by x\ and subtracting the second from the first, remembering equation (6), we have '^x/— 'Vyx' =. V {y'x"— x'y"). (24) Let r be the radius of curvature. Then we know that y'x" — x'y" ~ -. Hence we may Avrite OTHER METHODS OF VARIATIONS. 345 - — - {vxy' — Vyx') — {v^ COS A 4- Vy cos B\ (25) r V ^ where A is the angle which the normal makes with the axis of x^ and B the acute angle Avhich it makes with the axis of y. It is impossible to proceed further with the solution so long as the form of v is wholly undetermined ; but equation (25) will enable us to solve many problems with great ease, as we will now show. 282. Consider Prob. I. Here U=J^ ^ ds, so that v = i, Vx = o, % = o. Therefore equation (25) gives — = o. Hence, r being infinite, the solution must be a right line. Turn next to Case 2, Prob. II. Here ^^may be Avritten XSi (Is I I — —y SO that V =: -— =-, Vx = o, Vy = 3, and (25) - \y Vy V" prives — = and r = 21/ sec B. Let ;/ be the normal. Then 7t ^= y sec B and r = 2;/, which is known to indicate that the required curve must be a cycloid. In the last place, consider Prob. VII. Here we may write U— I 'yds, so that v = y, V:^ — o, Vy — i, and (25) gives t/So - = and r — — y sec B. Hence, in this case, the r y radius of curvature must equal the normal estimated in an opposite direction, and this is known to indicate that the curve is a catenary, the directrix being the axis of x. 283. In all these problems we shall obtain the same equa- tions at the limits for the determination of the arbitrary con- stants as we would if we had regarded x as the independent variable. For suppose, first, the curve is to connect two fixed points. Then, as shown in Art. 279, the hmiting terms 34^ CALCULUS OF VARLATIONS. Will take the form of (19), and v and / being always equal, they will entirely vanish, so that the constants must be determined by the circumstance that the curve is to pass through two fixed points, which are evidently the same conditions as we would have obtained had we assumed x as the independent variable. If we next require that the curve shall always have its extremities upon two fixed curves whose equations are as in Art. 278, then we shall obtain equations (21). Now the first of these equations gives no direct condition regarding the limits, but, with the aid of the second, serves merely to deter- mine ^ in (16), <; being an additional constant necessarily intro- duced by the employment of the new quantity /. But divid- ing the second of these equations by j/, and multiplying by \yx)v we find (i -^ f'yo^^ — o, and a similar equation for the lower limit. These equations show that the required curve must meet its limiting curves at right angles, which conditions are also the same as would have been obtained had we assumed X as the independent variable. Problem XLVI. 284. Lef V and u be any functions of x and y only, with con- stants, and let it be required to jnaximize and minimize the expres- sion Here, as before, because s has been made the independent variable, x and y, and consequently their variations, cannot be regarded as entirely independent. But equation (6), Art. 273, must always hold between x and j/; and as this gives an imphcit relation between them, the variation of that equation must involve such a relation between their variations. Hence, multiplying the variation of (6), as before, by an unknown OTHER METHODS OF VARIATIONS. 347 quantity /, and transforming the variations by equations (8) and (lo), we may, as before, write equation (ii). Art. 274. Now it will appear, by reasoning precisely like that em- ployed in the last problem, that to vary ^in the most general manner, even when the required curve is to pass through t\\ o fixed points, we must add to the terms at the limits the terms V^ds^ — V^ds^. For it is evident that the reasoning there used would be equally applicable if, instead of supposing £/ to be a function of x and y only, it had been any function of x, y, x', ^''\y,y'j etc. Now varying (i), adding equation (11), Art. 274, and integrating by parts as usual, we shall obtain 6U-^l\ ds, - V, ds^ + {u + lx'\ 6x, — {u + lx'\ dx, + / J//' dy^ — l.y^Sy^ + r^ \ \yx + u^x' - u'- {ix'y^Sx + {_vy -f uyx'- {iyy^dy\ds = L,-L, +£'\MSx + NSy\ds. ■ (3) Here, as before, L^ — L^ and the integral must severally van- ish whatever be the value of /. If now, as before, we suppose / to be such a quantity as will reduce M or N to zero throughout U, it will appear by the same reasoning as before that the other must vanish also. Making M and N zero in (3), we have (ix' -\-uY = v^-\- u^x', {lyy = Vy + tiyx\ (4) Multiplying these equations respectively by x^ and y, and adding, observing equation (6), Art. 273, we have I' -\- u'x' = v^x' -|- Vyy' -\- x'(uxx' -\- Uyy') = v' -\- u' x' . (5) Hence, as before, l' = v' and l=v-^c. (6) 34^ CALCULUS OF VARIATIONS. 285. Now in determining c we must remember, as before, that if we can express L^ — L^ in terms of ds^ and ds^, we may, since these quantities are independent, equate their coefficients severally to zero ; so that we need here consider but one limit. Let us first suppose that the curve is to pass through two. fixed points. Then, taking the value of L^ from (3), and substitut- ing in it the values of dx^ and dy^ from equations (17), Art. 277, and remembering equation (6), Art. 273, we find v^ — l^ = o, which shows, as before, that v = I throughout U, c in (6) being zero. Next suppose the curve is to connect two fixed curves whose equations are as in Art. 278. Then in L^ substitute the value of (^F, found by transposing equation (18), Art. 278, and equate the coefficients of ds^ and Sx^ severally to zero, because these quantities must be independent. Then, we shall have «. + /,-^-,' + hyj: = o, V, + u,x: + /j'.'^//.'- /,^." = o. (7) Multiplying the first of these equations by x^ and subtracting the second from the product, we shall, by observing equation (6), Art. 273, have l^ — i\ = o, so that here also v =^ I. 286. Putting v for /, and differentiating the first term in each, equations (4) become Vx — v'x' -\- Uxx' — u' ^= vx'\ Vy — v'y' -\- iiyx' =. vy" . (8) Now multiply the first term in each of these equations by x'' ^ y, and put for v' and u' their values. Then factoring, we have y^Oxy' — i^yx' — u,} = vx", x'ivyx' — v^y' + u,]) — vy" . (9) Multiplying the first of these equations by y\ the second by x\ and subtracting the second from the first, we readily ob- tain, as before, -— — - (z'x cos A -{- Vy cos B + ^y)' (10) OTHER METHODS OF VARIATIONS. 349 287. Let us next apply this formula to a few cases, begin- ning with Prob. XV. Here U —J^ {yji:' -\- a) ds, so that v = a, Vx = o, Vy = o, u =f, Uy= I. Therefore equation (lo) gives - z= . Hence the curve must be a circle, since r is a con- r a stant. The negative sign is in this case as it should be, be- cause it has been shown that a must be negative. Turn next to Prob. XVI. Here U =fj\/x' + ay)ds ; so that V = ay, v^ — o, Vy = a, it = y, Uy — 2y ; and equation (lo) gives • I _ cosB 2 r y a But = -, n being the normal; and as we have already y n shown that a must be negative, we may write — | — = — -. r 71 A We cannot in this case proceed to the solution obtained in Prob. XVI. without expressing the value of r and integrating as in that problem, although it is evident enough that the sphere will satisfy the last equation. We may remark, in passing, that Probs. XVII. and XVIII. are to be regarded as belonging to the preceding problem, because the factor of ds is a function of x and y only, together with constants. 288. Here also the conditions for the determination of the two constants which will enter the complete integral of equa- tion (id) will be always the same as though we had assumed X as the independent variable. For if the curve must pass through two fixed points, we shall have for the upper limit L, = {v, — /,)as, = (v, — v,)ds,. 350 CALCULUS OF VARLATIONS. That is, the Hmiting terms will vanish as they would by the other method. But suppose the curve is to connect two fixed curves. Then if x were the independent variable, we would obtain for the upper limit «Xi+/V+'''.(i+^///) = o; and multiplying by x\ remembering equation (6), Art. 273, we shall obtain the first of equations (7). Now the second of these equations gives no new condition, but merely enables us to determine the constant c in (6). To ascertain these con- ditions, let be the angle between the required and the upper fixed curve at their intersection, t the angle whose tangent is f\ and a the angle whose cosine is x' . Then, multiplying the first of equations (7) by cos /, we have u^co^ t -f- ^i(cos a cos t -f- sin a sin f) = u^cos t -\- v^co^ — o. (i i) Problem XLVII. 289, Let r be the radius of curvature of a plane curve ^ and V any function of r and constants. Then it is required to determine the conditions which will maximize or minimize the expression Here u=fyds. (I) SU=V,ds,- V, ds, +/'' VrSrds. (2) Now the following equations are known to be true : -^R= y'x" - x'y", \ = R' = x"' ^ y"\ r r \ (3) x" +/^ = I, x'6x' +ysy = o, x'x'' +yy'= o. J We must now obtain dr. We have ■ s{R') = 2{x"dx"+y'dy')=^ ^=^-. OTHER METHODS OF VARIATIONS. 351 Whence dr ^ - r\x"dx" -^y"^y"\ (4) Hence, proceeding as before, and putting v for F^r^ we have dU= V,ds,- V,ds, Ar r\- v{x"dx"-^y"Sy") + l{x'dx' ^ y'dy')\ds = o. (5) Whence, as usual, we obtain, after changing signs, the equa- tions iyx")" + {Ix')' = O, {vy")" + (//)' = o, (6) and (.^y y j^l^' ^a = vx'" + v'x" + lx\ I Multiplying the first of these equations by y' , the second by x' ^ and subtracting the second from the first, we have yi^yx'" - x'y'") + v\y'x" - x'y") = ay' - bx' = vR' + Rv' . (8) Whence „ ^^ 2 , , / x vR = Vrr^ = ay — bx -^ c. (9) 290. It will be seen that in this case / has been eliminated, and we will now examine the method of determining the con- stants in (9). Consider the terms at the upper limit, arising from the usual transformation of (5). These are VJs,+ \(vx'y-\-lx'\,Sx, + \{vyy + ly'\. Sy, - v,{x"dx' +y'Sy\ = o. (10) Now it at once appears from (7) that the coefficients of ^x^ and Sj/^ are respectively a and b ; and if for 6/ we put its value derived from the fourth of equations (3), the terms y beyond dy^ will become 352 CALCULUS OF VARIATLONS. Hence the terms at the upper limit become VJs,-\-adx,^bdy, - I ^- I dx: = o; (12) and a similar equation will evidently hold at the lower limit. Now the last term of the first member of (12) is evidently independent of the others, so that we must have Vrr" = o at both limits. Now suppose the line joining the extremities of the required curve be assumed as the axis of x. Then, because y and F^r^ vanish at both limits, we have, from (9), o =1 — bx^A^ c and o — — bx^-\- c\ so that b and c must vanish, and then (9) becomes VrT'^ay, (13) 291. Suppose the curve is to pass through two fixed points. Then the terms at the upper limit become V^ ds^ -\- aSx^ = (v — <^^0i ^^1 — ^' the second member resulting from the elimination of dx^ by means of equation (17), Art. 277 ; and a similar equation holds for the lower limit. But suppose the curve is to connect two fixed curves whose equations are as heretofore. Then the terms at the upper limit are Fj ds^ -[" ^^^1 + ^^Ji — ^' (14) Ehminating dj, by means of equation (18), Art. 278, and then equating severally to zero the coefficients of ds^ and Sx^, we shall obtain K + ¥/< - hy! = 0, a + bf: = 0; (15) and similar equations for the other limit. Now if the axis of X join the points of intersection of the required curve and the OTHER METHODS OF VARIATIONS. 353 two fixed curves, b will vanish, while a cannot, as appears from equation (13) ; so that the second of equations (15) can only be satisfied by supposing// to be infinite. Hence the tangents to the two fixed curves at their points of intersection with the required curve must be at right angles to the line joining those points. 292. As an example of the foregoing theory, consider Prob. III. Here SO that V=r, [>= i, and equation (13) gives r" = ay. Now as the axis of x in this case joins the two extremities of the required curve, it is readily seen that the cycloid having its cusps upon the axis of ;r is a solution, because in such a cycloid r = 2 VDy, D being the diameter of the generating circle. 293. Another interesting apphcation is the following: An elastic spring AB is adjusted between two right lines so as to be tangent to both at its extremities A and B ; it is required to determine the form which the spring must assume in order to be in equilibrium. According to the principle of Daniel Bernoulli, the curve AB must be such as to minimize the expression U =J --. J 2 Hence V = -, V,. — — r-> and equation (9) becomes r r = ay — bx -\- c, (16) 354 CALCULUS OF VARLATLONS. But since AB is compelled to be tangent to the lines AC and BD, its extreme tangents have a fixed inclination to the axis of X, and therefore d;r/, dj/, ^^J and Sj/J vanish, and we need not now have Vrr"" = o at either limit. But equations (15) are universally true, and the second of these gives a-^bf — o and a + bF'=^ o. (17) But since the lines A C and BD are not parallel, the constants /' and F' , which are the tangents of the inclinations of these lines to the axis of x, are unequal ; so that in this case we find that <^ and b must vanish. Then, by (16), we find that r is a constant, so that AB must be a circular arc if r be finite. But now the first of equations (15) would appear to give V — -2=0 for both limits ; which evidently cannot be true. To obviate this difficulty we must suppose the spring to have a given length. Then ds^ and ds^ will vanish, and the first of equations (15) will not necessarily hold. But under this supposition we should, according to Euler's method, have written V= — -{- d, which would produce no change in any equation except the first of equations (15) ; and this, when a and b vanish, would give- + <^'= o at either limit, which presents no difficulty. T/iird Method. 294. We have already seen that when x is the indepen- dent variable, we are, although the supposition is unnatural, permitted to vary x ; and in like manner, when s is the inde- pendent variable, we may ascribe variations to s throughout the range of integration. Indeed, this is the method usually adopted ; and as we are generally obliged to increase or de- crease s at its limits, the method does not seem altogether OTHER METHODS OF VARIATIONS. 355 unnatural. The following illustration may perhaps aid us in forming a better conception of the two methods. 295. Suppose we had a curve AB connecting two fixed points or two fixed curves, and suppose the curve to be formed of non-elastic wire on which notches are placed at our plea- sure, the wire extending somewhat beyond A and B. Then when we vary the form of AB in the most general manner consistent with variations, we shall, in general, find that we are unable to make the new curve connect the two points or curves without either adding or excluding certain wire ad- jacent to A and B. Still the distance of any notch from some given notch — that is, s — undergoes no change, a positive or negative increment merely being added to the limits. This may illustrate what takes place in the first method. Now suppose the original piece to be expanded by heat or contracted by cold until it is able to form the required arc of the derived curve. Then, although we increase or diminish the length of the arc AB, we do not add or exclude any wire. But now the distance of any notch from the given notch, or s, will have undergone an infinitesimal change ; that is, will have become s^ ds. But, to render the illustration complete, we must suppose the motion of any particular notch to be capable of taking either a positive or negative direction, or of becom- ing zero, or, in short, of following any law we please. In this 'case we would have an illustration of the method which we are now about to employ. 296. Let us now examine the mode of employing this method. Assume the equation U^£vds, (.) where V is any function of s, x, x' , x'\ . . . . y, y', y" . . . . Now when we vary s, x, y, etc., the reasoning in the begin- 356 CALCULUS OF VARIATLONS. ning of Art. 264 is rendered applicable to the present case by reading s for x. Moreover, all the equations, including (6), will be true if for x we substitute s in the limits, the differ- entials and the variations. Beginning then with (6), we have But t/so ds ^so ^ ' where accents denote total differential coefficients, while literal suffixes will denote partial differential coefficients ; so that V'=V,-^ V^x' + V^,x" + V^.x"' + etc. + Vyy + Vy.f + Vy.>y"' + CtC. (4) Now, to the first order, we have 6V^ V,ds-\- VJx+ V^.^x'+ V^nSx" -\-Qic. + Vy^y + Vy>dy + Vy..dy" + etc. (5) Hence + r \ Vx^^ + ^0"^^' + Va:'>dx" + etc. + Vy^y + Vy,sy^ Vy,.s/'+ etc. -( F^^'+ V^,x" + V^,.x"' + etc. + Vyy' + Vy,y" + F,„/"' + etc.) Ss\ds. (6) Now employing gd as before (Art. 265), let GD^ =z (^;ir — x'ds and cb?^ = 6y — y'Ss. OTHER METHODS OF VARIATIONS, 35/ Then, by the same method as that employed in Art. 265, we obtain Sx' = {G^y + x"Ss, dx'' = {G^y + x'^'Ss, etc., ) I (7) d/ = {ooyy +y'^s, dy" = {Goyy^ +/''^s, etc. ) But these equations are of course, Kke those in Art. 265, true to the first order only. By the use of these equations, (6) becomes + pi V,co- + VAoo^y + V,.{c^r + etc. + VyGDV + Vy^Goyy + VyioDvy + etc. \ ds. (8) Hence, by the usual transformation, and giving for brevity only the general form of the terms at the limits, we have 6U=^ Vds-\-{V^>- F^,/ + etc.)c^+(r^.,-etc.)(o^y + etc. + {Vy'- Vy>/ + etc.)c^^ + {Vy. -etc.)(G.^y + etc. . + {Vy- Vy/ + Vy>> " - ^tc:)ooy\ds. (9) 297. But dx and Sy, and consequently oof^ and coy, are not wholly independent, because, whether we vary s or not, the equations x'"" J^y:=i and x'x" + y'y" = (lo) must always hold throughout both the primitive and derived curve. If, therefore, we wish to maximize or minimize U, and for this purpose equate (^t/to zero, we must, as before, in order 358 CALCULUS OF VARIATIONS. to obtain any available equations of condition, employ the method of Lagrange. Now from (lo) we have x'dx' +y^/ = = x\g^)' + x'x"Ss-^y\Goy)' ^y'y"Ss = x\oo^)' -\-y\ojy)' 4- {x'x" +yy')^s = x'{GD^y -\-y{oDyy. (i i) Therefore, / being an undetermined quantity, we may, as be- fore, write £'i\x'{c^)'^y{oovy\ds Now transform this equation and add it to (9), and let L de- note the general form of the limiting terms L^ — Z^, M and N being the respective coefficients of ooP^ds and coyds under the integral sign. Then we shall have L^Vds-\-{ V^^ - V^>/+ etc. + Ix') G^ + ( V^u- etc.) (c^)'+ etc. + {Vv'- yy"'+ etc. + ly) ^y+ ( Vy. - etc.) {oovy + etc., (12) J/= F. - Vy + V,." - etc. - {IxJ, (13) N=Vy- vy + Vy." - etc. - {lyy. (14) Now it is evident that (13) and (14) are the same differen- tial equations as we would have obtained had we followed the preceding method, and ascribed no variation to s, I of course in each case being supposed to be so taken as to cause either M or N to vanish, so that the other will vanish also. Hence, since the general solution will have the same form as before, it will be necessary, in further comparing the two methods, to consider only the terms at the limits. 298. It may be observed, in the first place, that the gen- eral form of the limiting terms is the same by the two methods ; 6s^, 8s^ and the cos and their differential coefficients in the OTHER METHODS OF VARIATIONS. 359 second method replacing ds^, ds^ and the (^'s in the first. It would appear, therefore, that we might safely assume that the same conditions at the limits could be ultimately obtained by the two methods. But as it has not been deemed necessary to consider the most general form of V by the other method, it will, we presume, be sufficient to give Fthe same degree of generality in this ; that is, to show that in the three preceding problems the same equations at the limits are obtained by either method.. Suppose we make V a function of x and y only ; that is, apply this method to Prob. XLV. Then, by (12), we have, for the upper limit, L, = Vfis, + {lx'c^\ + (ly'c^y\ = 0. (15) Now suppose the curve is to pass through two fixed points. Then dx\ and S)\ vanish, because by this method x^ and y^ mean the co-ordinates of the actual extremities of the arc, al- though 6x^ need not vanish, as the arc may have undergone an alteration in length. Hence (0/^)^= — x^Ss^, {oofii)^— — yl^s^, and (15) gives Z,^|F-/(y^+yOh-o; (16) so that Fj = /,. ^ Next suppose the curve is to connect two fixed curves whose equations are as usual. In this case we shall have Sy^ =// ^x^. Substituting this value in (15) and equating severally to zero the coefficients of ^s^ and Sx^, because these quantities are entirely independent, that of Ss^ will give the second and third members of (i6), while that of dx^ will give (/y + //'/X = o. This is the same as the second of equations (21), Art. 279, the interpretation of which is given in Art. 283. 360 CALCULUS OF VARLATIONS. 299. Next consider Prob. XLVI. Here V — v -\- ux\ V and u being functions of x and y only, so that V^' = u. Therefore (12) gives {v + ux'),ds, + {u + lx'\{p^\ + hy;{oDy\ = o. (17) If now the curve is to pass through two fixed points, ^x^ and ^)/, and its coefficient must vanish ; so that, as before, b and c become zero, and we have VM + a{c^\ = o. (9) 362 CALCULUS OF VARLATIONS. Then if the extremities be fixed, d;ir, becomes zero, and we have, as before (Art. 291), (F— ax')^ = o. But if the extremi- ties are to be upon two given curves, then the terms at the limits become V,6s, + a{GD^), + d{c» Now it needs no additional explanation to show that if U is to become a minimum, the first integral in (2) must vanish, while the second must become invariably positive. Hence, to the first order, we have 6U=r'\ y' Sv'\. ^' Sz\dx = r' , ^^ Sy'dx + r^ ^' dz'dx = o. (3) But since z is also a function of x, we may put z, z\ z", etc., for y, /, y\ etc., in the reasoning of Art. 9. Then we shall d'^'^z find (^^^) =: -^— -. In like manner it is evident that when x SHORTEST CURVE IN SPACE. 365 receives no variation, if we had any number of variables y, z, u, etc., all regarded as functions of x, the reasoning of Art. 9 would apply to each, and we would have dy.,= ^^, tf^«)=f^-, /^-i\ d^jo(^-i), d^„ (^^0, .... (^^,(^-1), (^ir/"^-i), the number of which will be 2{m -^ n)-\-2, or merely 2{m -\- n), if the lim- iting values of x be fixed, or if dx^^ and dx^ be restricted as formerly. Moreover, it will appear, as before, that any con- dition which causes one of these equations to disappear will itself furnish a new equation of condition, so that the number of limiting equations will still remain equal to that of the arbitrary constants. Nevertheless it is easy to see that the reasoning here em- ployed may be subject to exceptions similar to those which have been explained in the case of two co-ordinates ; but these will give the reader no serious difficulty. 3(4. We may now consider, as being somewhat connected with our subject, the principle of least, or more properly mini- mum action, particular cases of which have been already dis- cussed. Problem L. A particle is to move in space from one fixed point to another, its motion being controlled solely by a system of incessant forces. Then x, y, and z being the co-ordinates of any point of its path, ds an element of tins path, and v the velocity of the particle at the end of any time t, it is required to show that the 7iature of this path must be such as to render S U to the first order zero, where Denoting by X, Y and Z, as usual, the aggregated com- ponents of all the forces in the direction of the axes of ^, j. 37^ CALCULUS OF VARLATLONS. and z respectively, we shall assume the well-known equation in mechanics, V dv = - div") = Xdx + Ydy + Zdz, (i) Now if we suppose the particle to be moving along the re- quired path, the symbol d, as applied to any quantity, denotes the change which that quantity undergoes when the indepen- dent variable, which we may here assume to be x, receives an infinitesimally small increment, the curve remaining unchanged. But if we draw any derived curve, and suppose the particle could pass from any point / on the primitive to some point P indefinitely near/, but on the derived curve, then if we give to the symbol d the meaning already explained, we may denote by 8 the corresponding change which the various quantities would undergo if the particle could pass from/ to P. Now in passing from / to P, just as in passing along any element of the primitive curve, we may assume that X, V, and Z remain constant ; and hence, denoting by ^^, v^ and v^ the components of v in the direction of x, y and z respectively, if we add to x, y, or z any infinitesimal increment, the corre- sponding change in ^ — -, etc., would be X, Y, or Z multiplied by those increments respectively, whether those increments were added as differentials for the purpose of enabhng us to pass from one point to another on the primitive curve, or as variations for the purpose of enabling us to pass from any point on the primitive curve to an adjacent point lying on some derived curve. But we have seen that any derived curve can be obtained without varying x, and we shall therefore consider/ and Pas having the same abscissa x. Hence, accents below denoting differentiation with respect to /, and those above with respect to X, and remembering that Y:=y^^ and Z — z^^, (i) may be written PROBLEM OF LEAST ACTLON. 377 i S iv") =v6v= Y6y + Zdz = fjj + ^,/.5'. (2) But the last member of (2) equals (x^7 + ^M^ - ky.^y^ + ^/^^.)- (3) Now we have 2 ds^ dx^ , dy , dz^ 21212 / N Then varying i)^ under the supposition that neither dt nor dx undergoes any change from variations, we shall obtain vSv=y^dy^^z^dz^. (5) That this supposition may be made will appear if we re- member that, in passing from p to P, v undergoes no change, so that dx and dt for that element of the curve maintain to each other whatever ratio they had before the curve was varied. Of course if we divide the whole time / into equal parts dt, the corresponding differentials of x cannot be sup- posed to be equal among themselves ; but this inequality can in no way affect our problem. Hence, admitting the validity of (5), equation (3) becomes {yfy + ^j^^)/ — "^ <^^, and (2) may therefore be written 2vdv = d{7>') =: {ySy + z^6z)^. (6) But since v — ---, we have v^ = ——. Hence (6) gives, after clearing fractions, d{vds) = d{yfy-\-zSz). (7) But since the particle is to pass from one fixed point to another, the derived curve must also pass through these 378 CALCULUS OF VARIATIONS, ' points, and we are not to suppose the particle capable of any displacement at either point, so that the variations of y and z vanish a1 both these points. Moreover, although we have really regarded t as the independent variable, we may inte- grate (7) as though that variable were x. For d in (7) denotes the change which y^dy-{-zfiz undergoes in the time dt, or while the particle passes from a point whose abscissa is x to one whose abscissa is x ^ dx\ so that it is the same thing whether we suppose these changes to be summed up for the time /j — t^ or through the distance x^ — x^. Therefore, by integration, (7) gives = {y,^y + ^/^). - {yfy + ^M- = o. (s) 316, To guard against certain misconceptions, we observe, first, that the reasoning here employed would not be applicable if the particle were compelled by a system of forces to move along a fixed material curve. For then, although equation (i) would hold, equation (2) would not, because that portion of X, Y and Z which arises from the normal pressure of the curve upon the particle would vanish for any point P without the curve ; so that we could not say, as formerly, that, in pass- ing from p to P, X, V, and Z would remain constant. We observe, secondly, that although the principle just established is commonly called that of least or minimum action, the name is not warranted, at least by the preceding demonstration. For our approximations were carried to the first order only ; so that we are merely able to say that the required curve must be such as to render (^f/to the first order zero. But we have already seen that the terms of the second order in (^^ do not always become positive, but sometimes vanish also, in which cases we inferred, although we did not investigate the matter, that .those of the third order would not likewise vanish, and that therefore d^ might have either sign SHORTEST LINE ON A SPHERE. 379 at our pleasure, thus showing that U could be neither a maxi- mum nor a minimum. It will be found, however, that the terms of the second order vn dU never become negative, and indeed it is generally conceded that the action can never, as Lagrange erroneously supposed, become a maximum. Section II. case in which the variations are connected by equations, differential or other. Problem LI. 316. It is required to determine the nature of the lifie of min- mum length which can be draivn between two fixed points or curves on the surface of a sphere. Here U =.C" VT+7'"=-+7^ dx =/;■ Vdx ; and taking the variation of U, and integrating in the usual manner, we have '^^» dx Vi+/' + z" ^'^ dx Vi-\-y" + z" = hfiy, - hfy, + HM - HM + r^MSydx + f'NSzdx = o. (i) 380 CALCULUS OF VARIATLONS. Now in this case the variations of y and z are not indepen- dent, all derived curves which cannot be drawn upon the sur- face of the given sphere being excluded from comparison with the primitive. Nevertheless it is evident that the integrated and the unintegrated parts of (^^must severally vanish. For we may suppose each part to be expressed in terms contain- ing one variation only, the other having been eliminated. Hence we have L,-L, = o, r\Mdy-\-N6z)dx^o, (2) But we have from the sphere x''+/-^z' = r\ ydy^zdz = 0, d^ = IlZ^» (3) Hence (2) may now be written X" 1 ^- ? 1 '^"^^ =iy''y<^^ = °- (4) Whence it will at once appear that to maximize or minimize U, M ' must vanish. Equating — M' to zero, we have d y' d z' y dx Vi+y + z'' dx Vi+y + z" z so that we obtain 1 d ---J-.= = i d-—J--= . (5) 317. Before proceeding, we shall find it necessary to change the mdependent variable to s. It is evident that (5) may be written I ,dy ' I ,dz ,^. -d-^- =z~d --. (6) y ds z ds SHORTEST LINE ON A SPHERE. 38 1 Although the symbol d in (6) denotes change incident upon changes in x, yet we were not originally bound to consider two consecutive values of dx as absolutely equal, and we may therefore suppose that these differentials were so taken as to make those of s always equal. Hence, regarding ds as always constant, multiplying by --, and denoting by accents differen- tiation with respect to ^, (6) becomes y Now multiply both the numerator and denominator of the first fraction by y' , and of the second by z' , and denote the A C resulting fractions by — and — , which are, of course, equal to each other and to the members of (7). Hence the quantities A, By C and D are in proportion, and therefore A + C:B + D::A : B::C'. Dwy" ly-z^' iz. Hence either member of (7) equals B-\-D yy+ zz' (8) But from the equation of the sphere, and also the equation x'"" -{- y -^ z''' — I, we have xx' + yy' + zz' = o, x'x" rf- y'y" + z'z" = o : and therefore (8) becomes — , so that we have X X" y" z" ^ = -J = -z' (9) 382 CALCULUS OF VARIATIONS. Now because (9) is true we may evidently write the following three equations : xd"^)! — yd^x = o, xd^^ — ^d^x =^ o, j/d^j2 — zd'^y = o. Integrating these equations by parts, we obtain xdy — ydx = a' ^ xdz — zdx = b' , ydz — zdy = c' . (10) Multiplying the first of these equations by z, the second by — r, the third by x, and adding the products, we shall obtain a'z — b'y + c'x — o. (11) In this equation the constants are infinitesimal, but dividing by one of them, as c' , the two resulting constants may have any value we please, infinitesimal, finite, or infinite ; and we may write X -\- ay -\- bz ^ o, the equation of a plane passing through the centre of the sphere. The required curve must, therefore, be a great circle. 3t8. To determine the constants a and b, we have, if we suppose the curve to connect two fixed points, the equations -^i + ^7i + ^-s", 3= o, x,-\-ay,-^bz,^o. But suppose the curve is to connect two given curves, and let the equations of the curve for the upper limit be y=f{x)=f, z=zF{x) = F, or dy=zfdx, dz^F'dx. ^12) Then we shall have, as in Art. 69, ^y. = {f' ~-y'\dx, and dz,={F' -z'\dx,, (13) Also it is evident, as before, that L, and L, must severally vanish, and L,= V{i-^y''-\-z'\dx, SHORTEST LINE ON A SPHERE. 383 Eliminating 8y^ and dz^ by means of (13), and reducing, we obtain i -{- y^ fl + z^ F^ = o, which shows that the great cir- cle must cut the limiting curve at right angles ; and a similar result can evidently be obtained for any curve at the lower limit. If we suppose the limiting values of x only to be fixed, we shall obtain for either limit, after having eliminated dz by (3), zdy ~ ydz = o = — c' , Hence (li) becomes a'z — b'y = o = z — a"y, or z — a"y, where a" remains undetermined, as it should, it .being the tan- gent of the inclination of the great circle to the plane of xy. We conclude, therefore, that the great circle must be so drawn that its intersection with the plane of xy shall always coin- cide with the axis of x, 319. It will appear by reference that U has here the same general form as in the first problem of this chapter ; and hence if the limiting values of x be fixed, the terms of the second order, as they at first arise, will be the same as in equation (11), Art. 307, which may be written dU—-J Sdx. But now the mode of eliminating Sz must be rendered exact to the second order, and for this purpose we have 6z = ^Uy + l^S/=-y-6y-l+^Sy\ dy -^^ 2df ^ z ^ 2z' ^' which will at once appear if we remember that x has no vari- ation, and that Sy and Sz are taken along any section of the sphere at right angles to the axis of x. Substituting this value of Sz, we shall evidently obtain, as the coefficient of Sy, M' , as before, which must be equated to zero as formerly. But since we must not reject the new term of the second order arising 3^4 CALCULUS OF VARLATIONS. from the elimination of dz, we add it to those already in the second, and the complete terms then become When the limiting values of x, y and z are fixed, and the arc joining the two fixed points is less than a semi-circumference, the sign of these terms is undoubtedly positive, as we know from other considerations that we have a minimum. Never- theless the author is unable to present any satisfactory general demonstration of the fact that these terms fulfil all the neces- sary conditions for a minimum. 320. The method employed in the preceding problem is not sufficiently general for all cases, since it is evident that the connecting equation may, as when s is the independent vari- able, be an unintegrable differential equation, which will not enable us to express dz in terms involving dy only. In this case we must adopt the method of Lagrange, with which the reader is already partially familiar, and which we will now briefly explain in a somewhat more general manner. 321. Suppose we seek to maximize or minimize the expres- sion U = I ^ Vdx, where V is any function of x, y, y\ . . . . z, z\ . . . .; and suppose also that the equation f{x, y, z, /, z' , ....) — o— f \s always to hold. Then, because/ is always zero, df must vanish ; that is, we must have fy^y + fy'^y + etc. +f,6z -\-A>Sz' + etc. = o. (i) Then / being any quantity, constant or variable, we may write / Idfdx = o. Now vary U to the first order, equate SU to t/ Xq zero, and transform by integration as usual. Then, in like METHOD OF LAGRANGE. 385 manner, transform / Udfdx, and add the result to dU. Then giving only the general form of the terms free from the inte- gral sign, we shall have 6U = \Vy, + lfy, - ( F,. + //,.)'+ etc. KJ. + etc. +{Vz'+ Ifz' - ( V,.. + If,.?)' + etc. \ Sz + etc. ^-rWv^-lfy - (F^ +//,-)' + etc.] tfj + Wz^ Ifz - {Vz'+^/z'Y + etc.] ^^}d;t: = o, (2) Now whatever be the value of /, the integrated and the unintegrated parts oi ^U must severally vanish. Then if we assume / so as to make the coefficients of either of the quan- tities ^j/dx or S^dx vanish, the coefficient of the other must vanish also. Thus we reduce (5^^ to such a form that, without eliminating either Sj/ or ^^, we may, in the unintegrated part, regard these quantities as if they were really independent, and equate their coefficients severally to zero. But it is evident that before we can obtain j/ and ^ as functions of x, we must be able either to eliminate / or to determine it also as a func- tion of X, y and s. This, however, can in general be accom- plished, because, in addition to the differential equations ob- tained by equating to zero the coefficients of 6y and Sz, we now have the equation /= o. 322. The arbitrary constants which enter the general solu- tion must evidently be determined by the conditions which are to hold at the limits. Denoting the terms at the limits by Zj — Zq, we may evidently in general equate these quantities severally to zero. Then in Z,, for example, the value which we have been obliged to assign to / will not usually cause the coefficients of dx\, Sy^, 6y^\ Sz^, Sz^\ etc., to vanish severally. Moreover, these variations are not independent, because the 386 CALCULUS OF VARLATIONS. equation /= o is to hold among them; so that we cannot equate these coefficients severally to zero. We see, therefore, that although we shall have as many equations at the limits as there are independent variations, a general discussion of the number of the arbitrary constants involved in the general solution, and of the number of the ancillary equations for their determination, must become com- plicated. Some results relative to these points have been obtained by Prof. Jellett, and are given in Art. 59 of his work, which results appear to be correct, although, as he himself states, they are at variance with some obtained by Poisson. It will here be sufficient to present Prof. Jellett's conclusions without demonstration. Let V be of the order n in y and m in ^, and let the con- necting equation / = o be of the order n' in y and m' in z, the limiting values of x only being fixed. Then, first supposing n ^ n' and in > ;;/, the number of constants involved in the general solution will be the greater of the two quantities 2{m -\- n') and 2(;// -|- 71), while the num- ber of the independent variations remaining in L^ — L^, whose coefficients may be equated to zero, will be the same ; so that all the constants can in this case be determined ; and the same conclusion holds when in > in' and n < n'. If we next suppose n < it' and in < in', the number of constants involved in the general solution will in general be 2{m' -\- n'), and of these constants there may remain undeter- mined any number not exceeding the lesser of the two quan- tities 2{in' — in) and 2{n' — n). 323. The method of Lagrange is capable of extension to any number of dependent variables. For example,, let V con- tain jr, y, z, u, and any differential coefficients of y, z and u with respect to x ; and let the equations fix, y, z, u, y', z', ii' , etc.) = o=f and F{x, y, z, u, y, z', ic', etc.) =0 = F METHOD OF LAGRANGE. 3^7 always hold. Then both d/and (^/^must vanish; and assum- ing /, as another undetermined quantity, we have r^lS/dx = o and r^LdEdx = o. Adding- both these equations to dV, and transforming as usual, the integrated and the unintegrated parts must severally van- ish. Then we may write r\MSy 4- N6:2 + Pdu) dx = 0\ and if we so assume / and /^ as to cause any two of the quan- tities M, N and P to vanish, the other must vanish also, after which / and /^ must as before be eliminated, or found as a function of x, y, z and ?/, in order that we may obtain a com- plete solution. 324. If we adopt the method of Lagrange in the preced- ing problem, we shall obtain the equations ly =r = o, Iz - = o, ' y' -0, Iz- d z' dx Vi+y+z'' dx Vi+y+z'' so that eliminating / we shall arrive at equation (6), Art. 3 1 7. Moreover, / will not, in this case, appear at all in the terms at the limits, which will therefore be of the same form as before. 325. The last example is merely an individual of an exten- sive class, and was discussed separately merely for the sake of introducing in a simple manner the subject of the present section. We shall now proceed to consider a very general problem given by Prof. Jellett, from the partial solution of which the last and many other examples can be readily solved. 3^^ CALCULUS OF VARIATIONS, Problem LII. Let V be any function of x, y and z, and let these quantities be also conjiected by the equation f{x, y, z)^^ o =: f Then it is re- quired to maximize or 7iiininiize the expression U^ r\Vi-\-y" + z''dx, Supposing y and z found as functions of x, we may evi- dently then regard x, y and z as the co-ordinates of some curve ; so that we may consider as usual that we require a curve whose co-ordinates shall be the values oi x, y and z in the general solution, and which we may therefore call the re- quired curve. Moreover, since the equation /=o may be regarded as the equation of a surface, we may suppose that the curve is required to lie upon this given surface. Now let ds be an element of this required curve. Then we may, as in the case of two co-ordinates, adopt s as the in- dependent variable, considering x, y and z as functions of s, which itself receives no variation. We must, however, in this case, adopt the method of Lagrange for three dependent variables. For, since s is to be the independent variable, x, y and z, besides satisfying the explicit equation /= o, must also satisfy the unintegrable differential equation dx"" . dy'' dz^ „ 326. Now transforming to s, we have CURVES ON SURFACES. 389 (2) dU = Z\ ds^ — V^ ds^ -[- / (^a;^-^ + "^y^y + ^2 ^^) ds — O, j(''V {x'dx' + ^'^j/' + ^'(^^0 ds = o, I/Sq where accents above denote differentiation with respect to s, and hteral sufhxes partial differentiation as hitherto. Then, proceeding as usual, we obtain ^^x + IJx - {Ix')' = O - 7;, + I J, - Ix"- X'l\ ""^y + ^Jy — Wy ^o = Vy-\- IJy - ly" - y'l', >■ 'Vz + /./. - (i^'y = = 2^. + Kfz - 1^" - ^'i'- . (3) Multiplying these equations by x', y' and z' respectively, and adding the products employing the equations fxx'-\-fyy'^f,z'^f = o, x'^+y'^ + z'^=ir v^ x' + Vyy' + v^ z' = v\ x'x" + y'y''' + z'z" =o\ (4) lxx'-\-lyy -^hz' =l\ ' ^ we have v' — I' = and l=v-{-a, (5) 327. Before proceeding, we must determine the constant a, and this will lead us to examine the terms at the hmits. These terms are v^ ds, — V, ds, -\- l,{x'dx -\-ySy + z'Sz\ - IXx'dx +y'Sy + z'S^\ = o. (6) Now if the required curve is to connect two fixed points, 390 CALCULUS OF VARIATIONS, we shall have, by reasoning like that employed in Arts. 276 and 2"]"], for either limit, Sx = — x'ds^ (Sj/ = — yds, and d^ =: — z'ds, (7) Kence, substituting these values in (6), and observing the second equation (i), we have {y — l)^ds^ — {y — l),ds^ — 0, so that v^=^ L * and I = v. But if the required curve is to connect two fixed curves, let the equations for the fixed curve at the upper limit be dy-=.pdx and dz=^qdx. Then, by reasoning precisely like that of Art. 278, we have ^y. +y^ds, = pSpx^ + x^ds^, ) \ (8) <^^j 4- z^ds^ = ^1(^-^1 + x^'ds^. ) Substituting in L^ the values of Sy^ and dz^ found from (8), we have, omitting sufhxes, V ds -{- Ix'dx -j- ly\p Sx -\- px'ds — y'ds) -\- lz\q 6x -\- qx'ds — z'ds) = o. (9) But ds^ and dx^ are entirely independent, so that, equating their coefficients severally to zero, we have V + ly'x'p — ly + Iz'x'q — Iz"" = o, ) (10) lx'^ly'p^lz'q=o.) Multiplying the second of these equations by^;ir', and subtract- ing from the first, observing the second of equations (i), we find, as before, that /, — %\ and /— v. CURVES ON SURFACES. 391 Now, supposing /^ or v^ not to vanish, divide the second of equations (lo) by /j^/, and we shall obtain which shows that the required curve must cut the fixed curve at right angles ; and a similar result can evidently be obtained for another fixed curve at the lower limit. 328. Let us now return to the general solution. Putting V for / in (3), we have '^x + IJx — 'VX" — X'v'=^ O, ^ '^y + ^Jy — "^y" — 7V= o, Vz + IJz — 'vz" — z'v' — o. (II) Let A, B and C be the angles made with the co-ordinate planes by the plane of that normal section which contains at any point the tangent to the required curve. Then, because the plane contains the normal to the surface, and also the tan- gent to the required curve, we must have the equations fx cos A -\-fy cos B -\-fz cos C = oA \ (12) x^ cos A -{- y cos B -\- z' cos C — o.) Hence, multiplying the first of equations (11) by cos^, the second by cos B, the third by cos C, and adding the products, we have Vx COS A -j- I'y COS B -\-Vz COS C — V ix" COS A ^ y" COS B -{- 2" COS C) = o. (13) Now let A^, B^ and C^ be the angles which r^, the radius ot curvature ot the required curve, makes with the co-ordinate axes. Then it is known that 392 CALCULUS OF VARIATIONS. COS A^ — — r^x" , cosB^ = — r^y\ cos C, = — r^z", (14) Equation (13) may therefore be written Vx COS A -|- %>y cos B-\-Vz cos C = (cos A cos A^ -\- cos B cos B^ -\- cos 6" cos C^). (i 5) ^/ Next let be the angle which the osculating plane to the curve makes with the plane of the aforesaid normal sections. Then it is known that cos A cos A ^ -\- cos B cos B^-\- cos C cos C^ = sin 0. (16) Also, r^^ being the radius of curvature of this normal section, we have, by Meunier's Theorem, sm' 0=1 '—. (17) Equation (15) may therefore be written under any one of these three forms : -^ , — -„- (^a; COSy4 + Vy COS B + ^^COS C)'\ r, r,r v sm r. {Vsc COS A -{- Vy COS B -\- VzCOS C), tan I . . , „ , ^. = [Vx COS A -{- Vy cos B + "^'^cos 6 ). [ (18) The first two equations are evident enough, but to obtam the third we have r; „ sm o -L^=zCOS'o=- —' r,/ tan o CURVES ON SURFACES. 393 Hence or Whence ^;: rj : : sin'' o : tan'' o^ sin' 'o:r; ::tanV:r,;. sin''^ tan"^ r: r,; 329. Equations (i8) are as far as we can carry the general solution, so long as the form of v is altogether undetermined ; but we may nevertheless deduce a remarkable property be- longing to this class of curves. Suppose we consider the integral U^=J — = / v^ds. Then, since dv^ I dv dv,_ I dv dv,_ I dv dx v' dx' dy v'dy' dz v'dz' the equation in v^, which will replace (15), will reduce to equa- tion (15) in V, except that the sign of the second member will be changed. Therefore the first of equations (18) includes the solution of both problems. Hence, because equation (15) in V can be converted into equation (15) m v^ by merely chang- ing the sign of r^, we conclude that if, upon a given surface, a curve whose equation is possess the property of rendenng / v ds 2i maximum or a minimum, the curve whose equation is r,— —f{x,y, z,y:c) — a maximum or a minimum. 394 CALCULUS OF VARIATIONS. 330. We may, before considering- particular cases, deduce another property of this class of curves. Let d be a fixed point and TT' a fixed curve, both being situated upon a given surface, and let the arc TT' be taken indefinitely small; then draw two curves OT smd 0T\ each having the property of maximizing or minimizing the expres- sion U— I vds, when the limits are fixed. Then, since each curve renders U a maximum or a minimum, they must both satisfy the same differential equations ; and unless we suppose that there could be two solutions, (9rand OT' must be indefi- nitely near at each point, so that OT' could be obtained by varying OT, and also the upper limiting value of x. Therefore OT'— (9 r must equal that portion only of (^t/ which is with- out the integral sign — that is, since the lower limit remains fixed, and /^ = v^ — must equal v^ ds + v^ x/dx, -4- V, yl^y^ + z\ z^'Sz^. (19) Denote TT' by dS. Then we must have dz, dz dz, = -77T- dS, — —r ds,. ' dS^ ds^ ^ Substituting these values in (19), we have OT' -OT=v,-v, {x"-\-y" + z'\ ds. CURVES ON SURFACES, 395 But denoting by t the angle OT'T, (20) gives OT- OT = v, cos tdS, (21) 331. From the second of equations (18) may be deduced two others, which will sometimes be found useful. Let A J J, B^^ and C^^ be the angles made by the osculating plane with the co-ordinate planes. Then we have cos A^^ = r, {z'y" - y'z"), cos B^^ = r^ {x'z" - ^'x"\ ) [ (22) cos C,, = r, {y'x" - x'y). ) Now let the equation of the given surface be ds = Zxdx -\- Zy dy. (23) Then we have Substituting this value in the second of equations (18), and then eliminating either x" or y" by the equation x'x''-^yy'^z'z"=o, we have either of the following forms : x"-{-z^z" = ^ ""^ ^ {v:c cos A-\-VyCOS>B-\- 2;;jCOs C)y, y"-^ZyZ- = (25) -^--tl^:^ {v^ cos A + Vy cos B + v^, cos C)x' . ■ 39^ CALCULUS OF VARIATIONS. 332. We may now proceed to consider some particular cases of the foregoing theory. Problem LIII. It is required to find the line of minimum length which can be drawn upon a given surface between two fixed points or two fixed curves situated upon the same surface. This problem is the simplest case of the preceding, to which it may be reduced by writing v =^ i, v^ =^ o, Vy ^=^ o and Vz = o. Whence the second members of equations (i8) must vanish, and the first of those equations will give r^ = r^^, which makes o vanish. We see, therefore, that curves of this class, or, as they are generally termed, geodesic or geodetic curves, must be such that their osculating plane at any point shall be perpendicular to the given surface at that point. Now, in Prob. LI., since the radius of every normal sec- tion of a sphere is the radius of the sphere itself, it at once appears that every geodesic curve drawn on its surface must be the arc of a great circle. We shall not, however, enter upon an extended discussion of geodesic lines, but shall give the chief points concerning them, following, as we have done since the beginning of Prob. LIL, the guidance mainly of Prof. Jellett. 333. The equation of a geodesic line is deducible in sev- eral ways. 1st. It is evident that all the reasoning in Prob. LI. would hold if the equation of the given surface were f — o, except that X, y and z in the denominators of the equations would be replaced hy fx, fj and fz respectively, the equations remain- ing otherwise unaltered. This would hold as far as equation (9), which would become X y z fx fy fz GEODESIC CURVES. 397 Hence we may evidently write the equations fx dy — fyd'^x = o, /p d'^ — f^d'^x = o, ) fyd'^ - /,dy = O. ) 2nd. If the equation of the given surface be d^ =1 Zxdx -\- Zy dy, then equations (25) give x" + z^. z" = o, /' + Zy z" = o. (2) 3rd. Or, because which, together with the equation dz — Zxdx — Zydy =z o, rep- resent the geodesic line. 334. As another property connected with these lines, we may notice that equation (21), Art. 330, will now become OT^ — OT— cos t dS. Now it is easily shown b}^ the differ- ential calculus that if the lines OT and OT' were right lines, the point O and the curve TT' being in free space, the above formula would hold. Therefore we infer that, so far as their 39^ CALCULUS OF VARIATIONS. change of length is concerned, we may, in the application of infinitesimals, regard all geodesic lines as right lines. But we must here note a difference between the right line and most other geodesic lines. For while the former is always the line of minimum length between two fixed points, the lat- ter are not in every case. For on a given surface suppose two indefinitely near geodesic lines to be drawn : these lines will in general intersect at some point. Here, then, we would haye between two points two indefinitely near geodesic curves satisfying the same differential equations, so that we would naturally infer, as in the case of two co-ordinates only, that the geodesic line ceases to be a minimum when the integral U ranges from one intersection to the other. This remark, although undoubtedly true, can only be called an inference, as we cannot apply Jacobi's Theorem with any success to this class of problems ; still the principle in question is well illus- trated in the case of the sphere, where the two geodesic curves will be two indefinitely near semi-circumferences, having of course the same length. Again, we have already seen that the curve which renders vds di maximum or a minimum must, if one or both ex- tremities are to be confined to fixed curves, cut its limiting curve at right angles, and hence this must be true also of geo- desic curves. If, therefore, from a fixed point upon a given surface geodesic curves of a given length be drawn in every direction, and the extremities, which are free, be joined by a curve, this curve must be of such a nature that at any point it may be at right angles to the geodesic curve drawn from the common point to that point. 335. We close the discussion of this subject by the con- sideration of one more particular geodesic curve, the discus- sion of which is not without interest, and appears to be due to Joachimstal. GEODESIC CURVES. 399 Problem LIV. // is required to determine the nature of the geodesic curve drawn upon the surface of a spheroid. Let the equation of the surface be ^ + ¥ + J^='' W Hence equations (2), Art. 333, become . :." = ^-, f=y^. (2) a z z Now let P denote the perpendicular from the centre upon the tangent plane to the surface at any point of the required curve, and D the semi-diameter of the spheroid drawn parallel to the tangent of the required curve through the same point. Then it is known that and P' a' '^ b'^ b' ^^ D'~ a'~^ b''^ b' ~ "''' (3) Now differentiating v, and putting for x^' and /' their values from (2), we have , 2b' Ixx' , yy' , zz\ „ bWz" But if we differentiate (i) twice, and in the result substitute for x" and y'^ their values from (2), observing also the values of u and v, we shall obtain bhiz" v= --. (5) 400 CALCULUS OF VARLATIONS. Dividing (4) by (5), we have v' u' — = , udv + vdu = o, {6\ V u ^ ^ whence uv is a constant, say c. It appears, therefore, from the first and third members of equations (3) that the geodesic curve must be of such a nature as will always render PD a constant, say c'"^, 336. We can also deduce another property of this curve. Let r be the radius of curvature. Then it is well known that for any curve in space we have 7 = ^'"+/"+^'". (7) Substituting in this equation for x""^ and y""^ their values from (2), and observing also the value of u, we have But from (5) we have so that I uz"''b' r" z" 3.V e P' z' ~ u'' - D^' I v' P' D' 7 r — P (8) (9) But we have PD — c'"^ and D^ = -p-^\ and therefore (9) gives We see, then, that the radius of curvature for the geodesic curve must vary inversely as the cube of the perpendicular BRACHISTOCHRONE UPON A GIVEN SURFACE. 4OI drawn from the centre of the spheroid to its tangent plane which touches the geodesic curve. 337. We now pass to another problem which is of con- siderable interest, following, as before, the guidance mainly of Prof. Jellett. Problem LV. A particle is coinpelled to move in a groove upon a given sur- face from one fixed point to another^ being urged by a system of forces which always re?ider Xdx -\- Ydy -\- Zdz a perfect dif- ferential. Then it is required to determine the nature of its path in order that it may move from the first point to the second in a minimtun ti^ne. Let t denote the time, V the tangential velocity, and ds an element of the required path. Then it is evident that we are to determine the curve which will minimize the expression But we have, by a well-known principle of mechanics, V = 2f{Xdx + Ydy -\-Zdz)= f{x, y, z). Hence we conclude that the present problem is merely an- other case of Prob. LI I., to which it may be reduced by writ- ing -p=^. But before we can employ any of the formulae obtained in that problem, we must also be able to determine the values of the partial differential coefficients of v with respect to x, y and 8* Now we have -d{V'):=VdV=Xdx-\-Ydy^Zdz, (i) 402 CALCULUS OF VARIATIONS. where the differentiation is total. But since F, and conse- quently F^ is a function of x, y and z, the partial differential of V^ with respect to any of these variables, as x, is the change which it would undergo if we could change x into X -\- dx, the other variables remaining unaltered, and this change is given in (i) by making dy and dz zero. Hence, de- noting partial differential coefficients, as before, by literal suffixes, we have -{V%= vv, = x, VVy = Y, VV,= Z.} (2) Now putting for v its value — , we find VV, X Y z (3) Substituting these values together with that of V in the first of equations (i8). Art. 328, we obtain V - (Xcos A + Fcos ^ + Zcos C)\ (4) 338. Although we cannot carry the solution any further while the problem retains its present general form, yet we can deduce some interesting properties belonging to this class of brachistochrone curves. Let R denote the resultant of the forces at any point of the required curve, the angle made by the osculating plane to the curve at that point with the plane of that normal section which contains the tangent to the required curve at that BRACHISTOCHRONE UPON A GIVEN SURFACE. 403 point, and O the angle which R makes with the perpendicular to the plane of this normal section erected at the aforesaid point. Then we know that -S^cos'^^, I-— ,==sm^, —-—^——~, (5) Now the aforesaid perpendicular to the plane of the normal section makes angles with the co-ordinate planes whose co- sines are numerically equal to cos^, cosB and cos C. Hence we see that (Xcos A + Fcos ^ + Zcos Cy = R' cos' 0, (6) Therefore we have sin'^ R'cos'O V'sm^o F'sin^ = R'qo^'0. = R cos O. (7) 339. It is evident that the members of the last equation may, so far as the preceding equations are concerned, have contrary signs, and we must therefore next justify our assump- tion that they should be taken alike. Now the pressure upon the curve in any direction is equal to the sum of the components, in that direction, of the result- ant and of the centrifugal force. Moreover, the total force at any point may be resolved into three : the first normal to the surface, which is destroyed by the surface ; the second along the tangent to the required curve, which tends to produce acceleration of motion ; and the third in the direction of that perpendicular which has been previously mentioned, and thic component would, if the particle were constrained to move in F' . a groove, cause pressure against its side. But —y is the cen- 404 CALCULUS OF VARIATIONS. trifugal force, and o is the complement of the angle made by r^ with the aforesaid perpendicular ; so that the members of the last equation equal numerically the respective components of the centrifugal force and the resultant in the direction of this perpendicular. Now if the components have contrary signs, then, since the pressure upon the side of the groove must equal their sum, it must become zero. But in this case no groove would be required ; the motion of the particle upon the surface being controlled solely by the given system of forces. But in accordance with the principle of minimum action, the path of the particle would, under the present sup- position, be that of minimum action upon the given surface with the given forces ; which is not the problem we wish now to discuss. The last of equations (7) is therefore correctly written for this case. 34-0. We see, then, from the last of equations (7), that the required curve must be such as to make the component of the centrifugal force perpendicular to the plane of that nor- mal section which contains the tangent to the required curve equal to the component of the resultant in the same direction. Again, we have Y^---^Rqq^0^2Rqo^G\ (8) which equation shows that if a particle urged by a system of forces move on a given surface in a groove of such a form as to render the time of passing from one fixed point to another a minimum, the pressure upon the side of the groove, when the particle is in motion, will be double what it would be if the particle were at that point held in a state of rest and still urged by the same forces. Again, if the resultant should lie in the plane of the afore- said normal section, cos O will vanish, and from (6) we shall have MINIMUM CURVE OF CONSTANT CURVATURE. 405 Xq,o^A^Yqo^B-\-Zqo^C—q, which, in (4), gives = o. Therefore the curve in this case must be a geodesic curve. 341. The following problem is also from the work of Prof. Jellett, and its complete solution appears to be due to him, although the problem itself had been previously discussed by Delaunay. Problem LVI. // is required to determijie the nature of the curve of umiiniuni length which can be drawn between two fixed points in free space, the radius of curvature of the curve being always an assigned constant. Let ds be an element of the required curve, and r the radius of curvature, which is a constant. Then, adopt- ing here also the arc as the independent variable, we are to determine the curve which will minimize the expression U — I ^ds. We have also, in order that we may be able to employ the method of Lagrange, the two additional equations :c'- +y^ + y^ = I, x'" +/'' + ^'" = ~ = R\ (i) Therefore, since the variation of W can give only the terms ds^ — ds^, it is easy to see that by following the method of Lagrange, / and /^ being two undetermined quantities, we shall obtain the equations {/,x")" - {/x'Y = o, {lyr - W)' = o. ) {i/r - (&')' = o. ) 406 CALCULUS OF VARIATIONS. Whence, by integration, (l^x")' -Ix' = a^ x"i; + I/" - Ix', {lyy - ly' - b = y"i; + ly" - ly, {i/y -iz' ^ c ^ z"i; + i^z'" - iz'. (3) Eliminating / between the first and second of these equations, we obtain {^y'jc" - x'y") i; + {y'x'" - x'y'") /, = ay' - bx'. This equation is immediately integrable, giving I J {/x" — x'y") = ay — bx -\- f. (4) In like manner, eliminating / between the third and first, and then between the second and third of these equations, and in- tegrating the tw^o resulting equations, we have l^ {x'z' — z'x") =^ ex — az +/,, ) \ (5) 342. Before proceeding further we must consider the mode of determining the constants in (4) and (5), and we begin by determining /^ and /. For this purpose, multiply the first of equations (3) by x'\ the second by y", the third by z", and add the products, observing that equations (i) hold, and that hence jc'x" + yy + z'z" = o and x"x'" + yy + z"z'" = o. (6) Then we have i^v/ = «y' + ^y' + r.--, R^/^^ax' + by + cz' + o-, (7) Differentiating the first of equations (6), and transposing, we have x'x'" + yy + z'z'" = - {x"' +y'' + z'") = - r\ (8) MINIMUM CURVE OF CONSTANT CURVATURE. 407 Now multiply the first of equations (3) by x' , the second by y, the third by z\ and add the products, observing the first of equations (i), and also equation (8). Then we obtain l^ - RH^-ax' - by' - cz'. (9) Hence, by the second of equations (7), we have l=.g-2RH, (10) 343. We must next consider the terms at the limits. Giving merely their general form, these are : Z = ^i- + 1 Ix' - (I/')' \dx-^\ly'- {lyy \ 6y -f ^ \ Iz' - {Iz")' \ dz j^l^S^:c"Sx'-^y"dy'^z"Sz'\^o. (11) Now suppose the extremities of the required curve to be fixed, but the extreme tangents to be wholly unrestricted. Then it is evident, first of all, that L^ and L^ must severally vanish. Now consider Z,, and take first those terms only which contain ds^, Sx^, dj/, and Sz^. Then, because the extreme points are fixed, we shall have, as usual, ^x^ — — x/ds^, Sy^ = — y/ds^, dz^ = — z/ds^, (12) which being substituted in that part of Zj, having first written will give, by employing the first of equations (i) and (6), and also equation (8), (i-/+R'/Xds,. (13) Now it is evident that we could, without restricting the extreme tangents, so vary the arc as to produce no change in its length, in which case ds^ would vanish, and the remaining 408 CALCULUS OF VARLATLONS. part of L^ would then vanish also. Hence we see that the two parts of L^ are independent, and we have L,ix"dx' -\-/dy' + z''dz\ = o. (14) As this equation can be satisfied by making either factor zero, let us suppose the second to vanish. Then, although Sx/, (5y/ and dz/ zltq not independent, we have, from the first of equa- tions (i), x'dx'+/dy + z'dj2' = 0; (15) and if, by this equation, we ehminate any one of the varia- tions, as ^z/, the two remaining variations may be regarded as independent, and their coefficients be equated severally to zero. Now in the second factor of (14) first eliminate (^^/, and equate to zero the coefficients of Sx/ and dj// ; then eliminate (5j//, and equate to zero those oi^x/ and <^z/. Then we shall ob- tain i^W - x'z"\ = o, {s'/' -y'B"), = o, {x'f -y'x"\ = o. (i6) If now we square these equations and add them, and then to the sum add the square of the first of equations (6), we shall obtain a result which may be written (^"+/"+o. (^"+y^+o, = ^^ (17) the last member following from equations (i). This would make the radms of curvature infinite at the upper limit; and as it is to have a constant value throughout [/, the required curve would become a right line. But if we reject this solu- tion and require that the radius of curvature shall have a con- stant finite value, the- second factor of (14) cannot vanish, and /,, must vanish. Now since the coefficient of ds^ in (13) must also vanish, we see that /, must become equal to unity. These values make ^ MINIMUM CURVE OF CONSTANT CURVATURE. 409 in equation (10) also unity, and the second of equations (7) be- comes l^FJ^ax'^by' -\-cz'^\, (18) 344. rt is evident that we can treat L^ in a similar man- ner, and shall obtain like results ; so that we have ^/i==o. ^,0 = 0, /, = 1, /o==i. (19) Now since the position of the origin is in our power, assume it at the lower limit. Then, since x^, y^ and z^ must vanish, we see at once from equations (4) and (5) that/,/ and/^ must severally vanish. Then, neglecting / / and /^, multi- ply equations (4) and (5) by z' , y' and x' respectively, and add the products. Then we shall find {ay — bx)z' -\- {ex — az)/ -j- (bz — cy)x' = o = a{yz' - zy') + b{zx' - xz') + c{xy' - yx'). (20) To integrate this equation, put zrj ior y, and xv ior z. Then (20) becomes du dv ail — b av — c so that, by integration, l{au — b) = l{av — c)-\- c^=. l{av — c) -\- Ic^^ = Ic^fyCiv — c). Now putting for u and v their values, removing the logarith- mic sign, and clearing fractions, we have ay — bx = c"{az — ex), (2ij which, being an equation of the first degree between three variables, is the equation of a plane, and, containmg no con- stant term, the plane passes through the origin. Now the 410 CALCULUS OF VARIATIONS. circle is the only plane curve of constant finite curvature, and this must therefore constitute the solution required. 345« But it is easy to see that the solution just obtained cannot always be applicable. For suppose the assigned value of r to be less than one half the line AB, A and B being the two fixed points. Then the circle whose radius is r cannot pass through both points, so that we are led to expect that if there can be any solution in such a case, it must be discon- tinuous. Now as no boundary presents itself along which the varia- tions of X, y and z are subject to any other restrictions than those which are imposed by equations (i), we infer that the discontinuous solution can consist only of some combination of arcs which satisfy equations (2), and consequently equations (3), which may be regarded as fundamental. Still it is evi- dent that we may, *as usual in cases of discontinuity, suppose a, b and c to have each different values for the different points of the discontinuous solution. But in the present case these constants cannot change their values. For let x^, y^ and z^ be the co-ordinates of the point in which two of the arcs which make up the discontinu- ous solution meet. Then the part of ^^ without the integral sign corresponding to this point, considered as being on the first arc, will involve only dx^, dy^, Sz^, dx^\ dy^ and dz^\ For it is only necessary to add the increment ds to the extreme limits s^ and s^, as the only reason why such increments are required is that we may obtain the privilege of varying the arc in the most general manner, which would require an in- crease or decrease in its length as a whole. Now the coeffi- cients of 6x^, dy^ and dz^ are the first members respectively of equations (3), with contrary signs; so that, denoting this part of dUhy Z2, we have L,=.- adx, - bdy, - cSz, + l^lx"dx' + y"Sy' + z"dz'\. (22) MINIMUM CURVE OF CONSTANT CURVATURE. 4II If now we denote by x^, y^ and z^ the co-ordinates of the same points considered as being upon the second arc, we shall have at that point, as in the case of two co-ordinates, the terms L^ — Zg, Z3 having the same form as L^. Hence these terms will not vanish unless a, b and c have the same values for each arc. 346. It appears, then, that the solution must consist of some combination of circular arcs, all having the radius r, and situated in the same plane. But l^ must vanish at the ex- treme limits ; and we see from (22). that to make L^ — L^ vanish, we must also make /^^ and l^^ severally vanish, or must have ■^/ = -^V> 7/= /a'. ^/^-S"/, /,, = /,3. (23) We see, also, from (18), that when the first three of equa- tions (23) are satisfied, the last will be satisfied also ; so that we infer that the arcs are also to be so placed as to have a common tangent at their point of meeting, unless, indeed, we can make l^^ and l^^ vanish without such a construction. Now since a, b and c are unchangeable throughout the integral, (4) and (5), Avhich are derived from the fundamental equations (3), must also hold, as must equation (18); and as the arcs must lie in one plane, we need no longer employ three co-ordinates. Assuming, therefore, the plane of the arcs as that of xy, make z and its differentials zero. Then, because l^ must vanish at both extreme limits, while x^ = o and y^ = o, it is clear that/, /, /^ and c must vanish, so that equations (4), (5) and (18) become respectively l^iy'x" - x'y") = ay - bx, l^ = r\i -\- ax' + by'), (24) 34-7, It appears, then, that arcs of the same circle, so joined as to have a common tangent, will give at least one solution of the problem, provided equations (24) are satisfied throughout the entire range of the integration ; and this point we next proceed to consider. 412 CALCULUS OF VARIATIONS. "Let A and B be the two fixed points, and suppose we take three arcs, A CD, DEF and FGB ; and moreover, since the ori- gin only is fixed, it being at A, let the axis of x take the direc- tion AB. E ^ B Now taking first the arc A CD, its general equation may be written (x~hyj^{y-kf=r'- (25) where h and k are the co-ordinates of the centre ; and Ave shall suppose X and y to be so estimated as to render these co- ordinates positive. Differentiating (25), we have {x - h)x' + (7 - k)y' = o. (26) Substituting from (26) in the first of equations (i), we easily find ^' ■=±{y- k)R, y =^:{x- k)R. (27) Now if we suppose x and s to increase together, x' is always positive ; and y ~ k being negative, we must take the nega- tive sign. But the arc being below x, y' will be positive or negative according as x - h is positive or negative; so that for It we take the positive sign. We therefore have for this arc x'-^- R{y _ k), / = R{x - h\ \ X" =.-Ry'^_ T^Y^r ~h\ y" =: Rx' = _ R\y _ k\ ) ^^^^ Substituting these values in equations (24), they become - Rl^ ^ay~ bx, l^ = r'\i+a{y~ k)R + b{x - h)R}. (29) MINIMUM CURVE OF CONSTANT CURVATURE. 413 Substituting in the first of these equations the value of l^ from the second, we obtain r -\- ak — bh ^^ o. (30) Now consider the arc DBF, and let H and K be the co- ordinates of its centre. Then, proceeding as before, we shall find that we must now reverse the signs of ^ and y, which will leave those of x^^ and y unchanged, and equations (24) will become ie/, = aj- dx, /^ = r'\i+ a{y - K) R - b{x - H)R\, Whence we obtain, as before, r-aK+bH = o. (31) But since the arcs have the same radius, and a common tan- gent at D, we must have K =z — k] so that (30) and (31) give b{h -\- H) — o, an impossible equation unless b vanish. Under this supposition, the second of equations (24) becomes l,^r'{l-{-ax'). (32) But x\ being always positive, has the same value at D as at H, and therefore, since /^ must vanish at the latter point, it will vanish at the former also. If, on the other hand, we had required for the ^ltcACB the conditions which would cause /, to vanish at D, as well as at A, we would have found it necessary to make b vanish, be- cause, while the value of x' is the same at both points, those of y are numerically equal, but have contrary signs, and therefore the second of equations (24) could not otherwise be satisfied. Then equations (30) and (31) would become r-\-ak = and r — aK = o, so that K = — /b, a.s before. It appears, moreover, from (32), that if /^ vanish at A and D, it will also vanish at F; and that if we had taken any num- 414 CALCULUS OF VARIATIONS. ber of arcs, instead of three, /^ would vanish at each point of junction. 348. We see, then, that the proposed system of arcs not only gives a solution which satisfies equations (23), but it is also that which is necessary in order that /^ may vanish at each point at which discontinuity occurs, so that we have no reason to expect any other solution. But as we may take as many arcs as we please, all having the assigned radius, it is evident that we can make the system differ practically in no respect from a right line, which was a former solution. 349, We have thus far supposed that the curve is to be drawn between two fixed points, but let us next require its extremities to be confined to two surfaces whose equations are v ^ o and F= o, z* and V being functions of x^ y and z only. Then, considering the upper limit, we see that Z, be- comes, by the aid of (3), L^-= ds- aSx- bSy- cdz,-\- l^,{^"dx'-\-y'dy-\-z''dz'),=^ o; (33) and since /^^ vanishes, we have Zj = ds^ — aSx^ — bSy^ — cSz^ = o. (34) Now let X, + [_Sx,\ y^ + \6y;\ and z, + [dz;\ be the co- ordinates of the point in w^hich the required curve, after having been varied, meets the surface. Then we have, omit- ing the suffix i, v,iSx-\ + Vy[dy-] + v,[_Sz-] = O. (35) But we have in this case [dx] =dx + x'ds, [(5>] = 6> + yds, {Szl = dz-\-z' ds ; 30 that (35) becomes {v^dx + Vy Sy + 7'^ dz\ + {y^ x' J^Vyy' + v^ z'\ ds, = O. (36) MINIMUM CURVE OF CONSTANT CURVATURE. 415 Substituting the value of ds^ from (34), we obtain + V^y + b {vxx' + Vyf + v^ ^OK^Ji + S^^ -\-ciyxx' + e/^y + v^ z')\fiz, = o. (37) We may now regard Sx^, dy^ and Sz^ as independent, and may therefore equate their coefficients severally to zero. Per- forming this operation, we easily deduce Vx a c (38) and a similar equation in V evidently holds for the lower limit. Now from equations (4) and (5), since /^ vanishes at both lim- its, and /, fj and f^^ are zero, we have ay^ — bx^ = o, cx^ — az^ = o, dz, — cy, = o. ay, — bx, = o, 1 ex, — az, — o bz, — ^fo = o- J (39) Whence, by subtraction, we deduce X. — X^ Therefore, from (38), we obtain i—^—] = ( ^^ ] = ( ^^ ] > (40) (41) and a similar equation in Ffor the lower limit. These equations show that the straight line joining the ex- tremities of the arc must be normal to the two given surfaces. 4l6 CALCULUS OF VARL4TI0NS. 350. We have hitherto supposed the extreme tangents of the required curve to be wholly unrestricted ; but if we re- quire these tangents to have certain assigned directions, it is evident that the preceding figure cannot always give the gen- eral solution of the problem, since these tangents might be so assigned as not to lie in the same plane. It is shown in the following manner by Prof. Todhunter, in his History of Variations, Art. 156, that the solution in such cases will sometimes be a helix. The discontinuous solution will be found in Art. 154. Since dx' , Sy' and dz' are zero at both limits, it is no longer certain that /^ will vanish at either limit. Let us suppose, however, that the conditions relative to the limits are such that in equations (4) and (5), a, b,f^ and f^^ vanish. Then the second of equations (7) becomes RH, = cz'-irg. (42) i\lso, the terms at the upper limit will now become Zj = ds^ — cSz^ = o ; and the extremities of the curve being fixed, dz^ = — z/ds^ ; so that we have I + cz/ = o. (43) But Z, also gives rise to equation (13), so that XV, - /,) = «,'. Hence we see from (42) that^= /^ and from (10) that Z^, van. ishes, and then from (13) that /^ = i =z o-; so that (42) becomes . ' /^ = rXi+czy (44> Now assume x = h cos v, y ^^ h sin v, and z = kv. Then we easily obtain the following equations: MINIMUM CURVE OF CONSTANT CURVATURE. 417 ds - _ \/lf^^, x' == ^_- y= Vh'-^-M' Vh'-\-k' — X ,2 I /.a' -^ 7.2 From (46) and the second of equations (i) we find Hence (44) becomes ck and since a, b, f^ and f^^ are zero, equation (4) becomes ck ) je ^Ni+-^ while either of equations (5) gives ck ) k /. r^^i c. (45) {46) (47) (48) Substituting for r' in (49) its value, that equation becomes (49) SO that we have Whence k VW^^ = c{h' -^ k') ' rip' VW^' = %-ck. k c ^ VW+J ' k h'-kf (50) (51) 41 8 'CALCULUS OF VARIATIONS. Next substituting the value of r^ in (48), it becomes -f=V/i -^k +ck=-^ = If —^r^' (52) From equations (50) and (52) we see that we cannot have h and k equal ; but with this exception the assumptions X =: h cos Vy y =: h sin V and z ^^ kv will satisfy all the con- ditions of the question, and the helix will therefore be the solution required. 351. When problems of relative maj^ima or minima are to be considered, the same method must be adopted as in the case of two co-ordinates ; that is, we multiply the integral which is to remain constant bj' a constant, say a ; and it seems, therefore, unnecessary to introduce here any question of this class. Indeed, as the method of treatmg all the problems which belong to this section, whether of absolute or relative maxima and minima, is quite uniform, our knowledge of the calculus of variations would not be materially increased by their multiplication. Moreover, these questions generally lead us into work of considerable length, and rarely afford us any solution in finite terms, and are therefore somewhat wearisome. We shall therefore merely state two or three additional problems which the reader will find in the work by Prof. Jellett, or in the more recent French work, Calcul des Variations, by Moigno and Lindelof. (i) To draw between two fixed points or curves upon a given surface a curve which will maximize or minimize the expression Ur^ r\v-^Vx')ds, V and V being functions of the co-ordinates x, y and z only. (2) Two fixed pomts on a surface being given, and a curve connecting them, it is required to draw between these points a curve of given length such that the portion of the given GENERAL REMARKS. 419 surface included between the given and the required curve may be a maximum. (3) To find the form which a cord resting upon a given surface must assume in order that its centre of gravity may be as low as possible. 362. It will readily appear that while the adoption of s as the independent variable often presents great advantages in the discussion of the terms of the first order, it is exceedingly unfavorable to a successful examination of those of the second order. For, in the first place, even when the limiting values of X, y, 2, etc., are fixed, we would be obliged to add to SU the terms and then the relations between ds and 6x, 6y and S2 at either limit, which we have previously used, and which are true to the first order only, must be replaced by more accurate equa- tions. In the use of these more accurate equations, certain terms of the second order will evidently arise at the limits ; and as we may only equate those of the first order to zero, these terms cannot be neglected, but must be added to those already in the second order, thus rendering them more com- plicated. In the second place, when we are obliged to use the method of Lagrange, we must render that method true to the second order, which we have not hitherto done. To accom^ plish this, w^hether ;i: or ^ be the independent variable, we first take the variation of U to the second order. Then, supposing the connecting equation to ht f{x,y, ^) = o =/, we shall have ^f = fx^^ + fy^y -\- fz^2 + t(/-^^' + ^fxy^^^y + fyy^/ + etc.) =: O. 420 CALCULUS OF VARIATIONS. Hence we may write // Sfdx = o, where the Umits depend upon the independent variable. Now after having given to / such a value as will cause the terms of the first order to vanish after // d/dx has been added to ^U, we must remember that the variations in the terms of the second order are not independent, but are still connected by the equation /= o. If then these terms should be certainly invariably positive or negative, we have a minimum in the former and a maxi- mum in the latter case. But as we shall generally be unable, if /be a differential equation, to impose this restriction in any explicit manner upon the variations, we shall not usually be successful in determining the sign of these terms. Of course when / is a differential equation, its variation is taken to the second order, as already explained. 363. In the discussion of problems involving three co- ordinates, we have, according to our usual method, ascribed no variation to the independent variable, whether that vari- able be X or s. But it is quite common among writers to vary the independent variable also, just as has been already ex- plained for problems of two co-ordinates. Consider first, for a moment, the case in which ;r is the independent variable. Here we follow without change the reasoning of Art. 264 until we arrive at equation (7), after which we still follow the article, only observing, in finding the values of -^ and dF, that V is now a function of x,^',/, . . . . z, z' , . . . . Then, having obtained the longer expression for SU, which will replace equation (10), it is evident that equations (15), Art. 265, will still hold true to the first order, and that, by the same reasoning for x and z, we shall have the additional equations GENERAL REMARKS. 4^1 (^^^ = ^ + z" Sx, dz" = ^^ + z'^'dx, etc. ; (i) dx ax- ^ ' where cso^ ^^ Sz — z'Sx. Therefore, proceeding as in Art. 266, we shall obtain, instead of equation (17), an expression for dU identical in form with that which would result from ascribing no variation to x, except that oo and 00^ will replace dy and ^z. Moreover, since, to the first order, gd and cs^ equal dy and dz in the other method, we see that whatever relations may hold between these variations when the ordinary method is employed must hold also between od and (^ when x is varied, so that, as in the case of two co-ordinates, the same general equations will be obtained by either method, and it will be found also that the same equations at the limits can be estab- lished by either method. Next, when s is the independent variable, we proceed as in Art. 296, merely observing, in finding the values of v' and Sv^ that V is now a function of s and x, y and z with their differen- tial coefficients with respect to s. Moreover, we shall have, in addition to equations (7) of that article, which will still hold true to the first order, the equations dz' = (GD^y + z"Ss, 6z" = {oD^y + z'^'Ss, etc., (2) where gd^ r= Sz — z'Ss. Hence, as in the case of two co-ordinates only, we shall find that d^will take the same form as if we had ascribed no variations to s, except that go^, gdv and gd^ will take the place of 6xj 6y and dz respectively. CHAPTER III. MAXIMA AND MINIMA OF MULTIPLE INTEGRALS. Section I. CASE IN WHICH U IS A DOUBLE INTEGRAL; THE LIMITING VALUES OF X, V, Z, ETC, BEING FIXED. Problem LVII. 354. Suppose we require the form of the surface of least area terminated in all directions by a certain fixed and closed linear boundary. If this boundary were a plane curve or any linear figure situated entirely in the same plane, the required surface would of course be itself plane. But we here wish that the bound- ing frame or edge may have any assigned form whatever, not inconsistent with the condition that it shall be closed. Suppose, then, the required minimum surface to have been obtained, and call it the required surface, and suppose we take any other surface having a common edge with the first, and call this the derived surface. Then it will appear, as in Prob. I., that to prove the required surface to be that of least area we must, in the first place, assume that the derived surface with which its area is compared differs from it in form infinitesimally only. Then if the surface found have a less area than any such derived surface, it will be a SURFACE OF MINIMUM AREA. 423 minimum, that term being used in the technical sense already explained, and it will then be in order, in discussing the least surface, to consider whether there may be any other minima. We shall then at present discuss only the problem of find- ing the minimum surface. 356. Now let x, y and z be the co-ordinates of any point of the required surface, and suppose four indefinite planes — two parallel to the plane of yz, and two parallel to that of xz, the distance between the former two being dx, and between the latter two dy. Then, denoting by ds an element of the surface intercepted at any point by any four planes drawn as above, it is known that we shall have ds^Vi -^z"-^z;dydx, where accents above denote total differentiation with respect to Xy and those below the same with respect to y. See De Morgan's Diff. and Integ. Calc, p. 444. Therefore, designating the surface whose area is to become a minimum by U, we have to minimize the expression U= / Vi^z'-'-^z'dydx^ / / Vdydx, (i) 356. It is essential that we should here recall from the theory of double integration a clear conception of the precise meaning of equation (i). Suppose, then, the entire surface to be divided into strips by planes parallel to that of yz, the distance between these planes being dx. Then, the area in question equals the sum of these strips, while that of any strip is itself equal to the sum of the elemental areas intercepted on it by successive planes parallel to that of xz, and sepa- rated by the distance dy. To effect this latter summation, which we shall always suppose to have been first accomplished, we must imagme 424 CALCULUS OF VARIATIONS. the value of V to have been obtained from the general equa- tion of the surface, thus rendering Fsome function of x and y only, since z is some function of x and y ; and then, as x and dx will have the same value for every element of the same strip, while y will vary, we must integrate the expression Vdydx under the supposition that x and dx are constants. But since the summation must extend throughout any strip which we wish to consider, if we denote by y^ and y^ the values of y at its extremities, the area of any strip will evi- dently be given by the expression / Vdydx, x and dx being treated as constants. But because V was made a function of X and J ov\j,J Vdy will be a function of these quantities ; and since for any particular strip y^ and y^ will certainly be func- tions of X only, and perhaps constants, if we put 5 for the area of any strip, we may write 5 = f{x) dx = fdx, (2) Now to effect the summation of the strips, which is always the latter process, we suppose the edge or contour of the sur- face, when it has been projected upon the plane of xy, to form a curve capable of being expressed by the equation y — Fix), which curve we shall call Xh^ projected contour. Then equation (2), which was before true for any strip, becomes so for every strip. Hence we need no longer regard x as con- stant ; and integrating from x^ to x^, where x^ and x^ denote the extreme abscissas of the surface, or rather of the projected contour, we shall obtain the entire surface U. 357. Hitherto we have usually employed the suffixes o, i, etc., to denote* what the quantity to which they are applied will become when the independent variable receives a partic- ular value. Now because, in the discussion of curves, whether situated in space or not, we have but one independent vari- SURFACE OF MINIMUM AREA. 425 able, ;ir or J or some other, this method is satisfactory. But in problems relative to surfaces, where no curve is traced, x and y are evidently entirely independent, so that the substitution of a particular value of one variable does not necessitate the substitution of any particular value of the other, as it would if we were discussing a curve. It is, therefore, important that we should be able to specify just what substitutions of each variable are to be made in any function, which cannot be con- veniently done by suffixes, particularly when we come to in- tegrals of the third or higher order, involving three or more independent variables. These substitutions are indicated in the following simple manner. Let x, y, z, etc., be any quantities whatever, and let F be any function of these quantities. Then when we put for any of these quantities a particular value, as x^ for x, we write i% it being always understood that a suffixed quantity is sub- stituted for the unsuffixed one of the same name. Also, if it be necessary to denote that x^ has been substituted for x and y^ for y, we write the new function thus, I F^ where we shall always suppose y^ to have been first substituted for y, after which x^ is substituted for x in the resulting function. Again, suppose / to be a function integrable with respect to X, its general integral being F, Then we may write J.Jdx = F,-F,= l F- I F Now as we shall often have expressions of a similar form aris- ing from definite integration, we write the last equation thus, / fdx = /^ F, where it will be always signified that we are to substitute successivel}^ the upper and lower suffixed for the corresponding unsuffixed quantity, and then subtract the sec- ond result from the first. 426 CALCULUS OF VARIATIONS. Extending this principle still further, j^ j F will denote the following operations : tirst, that we must substitute suc- cessively j^i and Jo for jK, and subtract the second result from the first ; and second, that in the result we must substitute x^ and x^ for x^ and subtract as before. Thus we shall have F= \ F- F Ixq lyo Ixq ) I / /iCi / ?/i Ix^ ly^ Ixa lyi /a-o /2/o = 1 I F~l I P-l I F^l I F- (3) 358. The idea of employing a sign to denote substitution is due to M. Sarrus, who calls it the sign pf substitution, a name which we shall retain ; and it seems probable that, as Prof. Todhunter has remarked, since Lagrange introduced his symbol ^, nothing has been suggested which, is of such service to the calculus of variations as this sign. But the sign and the method of employing it were subsequently modified by Cauchy, whose method we substantially follow, as exhibited in the Calcid des Variations by Moigno and Lindelof. 359. As an illustration of the preceding discussion, let us suppose the given surface to be spherical, taking the origin at its centre, and considering only some portion of the upper hemisphere, whose edge or contour is to have any form we please. We may notice that z' is the partial differential coef- ficient of z with respect to x, and is obtained therefore from the equation of the surface by regarding y as constant ; and similarly x must be constant in obtaining z^. The equation of the sphere is x" ^ y" -^ z" = r". Whence z' = X — X -I 2 ^r'-x'-f z Vi+z- + z;=---^^—r—: Vr'-x"'-/ SURFACE OF MINIMUM AREA, 427 SO that (i) becomes jj __ n^x rvi rdydx ~ tAo Jyo vp — x^ —y But regarding x as constant, we have /-— ^-^^rsin- J^^ + c: and the definite integral may be written S = / rsm-' — ^ 'yo |/^. _ ^. dx. Thus we see that 5 does not contain z, and is also indepen- dent of the general values of y, but may still be some function of X. Now if we wish to denote the area of any particular strip for which x = Xa, we have only to write / / r sm - — -^ dx. I Iv^ ^/r" - x' To complete the integration, let us require all the surface for which neither x, y nor 2 shall become negative. Then we shall have y. = 0, y, = Vr' — x\ S= , U=: / = / ; and since the entire eighth of the sphere is required, Xa= o and x^ = r, and [/ — 2 360. Returning now to our original problem, we see that we can pass from any given surface to any other differing from it infinitesimally in form, and having a common edge, by 428 CALCULUS OF VARIATIONS. giving to z suitable infinitesimal increments throughout the surface, the values of both x and y undergoing no change ; and as dz indicates the change which z undergoes when we pass from one point to its consecutive on the same surface, we des- ignate the new increments, as before, by Sz. Moreover, we can also, without varying x or jj/, obtain the derived surface by giving infinitesimal variations to z' and z^, which are the tan- gents of the angles made with the plane of xy by those two edges of any elemental area which meet at the point x^ y, z. If now we denote by (^^ the change in area which the en- tire surface will undergo when z, z' and z^ receive infinitesimal variations, the required surface must evidently be such as to render df/ negative. But as we cannot express ^in any more explicit form than that given in (i), and as we must compare the required surface with such as can be derived by infinites- imal changes in its form, Ave are compelled to seek the varia- tion of the double integral in (i) in order to determine what conditions will render the variation negative. 361. In order to consider the subject more generally, let us assume the equation where Fis any function of Xy y, z, z' and z^, the limiting values of X, y and z being fixed ; and let us, for convenience, write z' =/, ^^ = q^ z" = r, zl — s and z^^ = /. Then if we change z into z -\-'^z, p into p -\- ^p and g into g + Sg, x, dx, y and dy remaining unaltered, and denote hy SU and dV the changes which U and V will undergo, we shall have = rrVdydx-\-rr,Vdydx. SURFACE OF MINIMUM AREA, 429 Whence, from (i), We have now merely to determine (^Fby Taylor's Theorem, which, since x and y undergo no change, will give + Vy^sp'^ 2 1\ Sz 6q + 2 Fp, Sp 6q + V^^ Sq^ [ + etc. ; (3) where the etc. denotes terms of the third and higher orders, and the differentials of V are all partial. 362. Now denoting by vS the terms of the second order in S V, with the exception of the -, we shall have ^^= £?fy''^ ^''^'+ V,,SpJ^V^Sq\dydx + 2X. X Sdydx-^^^o. (4) If now we require that U shall become either a maximum or a minimum, it will, since Sz^ Sp and 6q are entirely in our power and may have either sign, appear, by precisely the same reasoning as in the case of single integrals, that the first integral in (4) must vanish, while the second must become invariably positive for a minimum and negative for a maximum. Now we must observe that x and y are completely inde- pendent, and that z' and z" or / and r are the differential co- efficients of z with respect to x, y being regarded as constant ; 430 CALCULUS OF VARLATIONS. that is, in finding them, we regard -S' as a function of x only, and constants, among which we reckon y. Or we may regard z as the ordinate of the curve made by the intersection of the required surface with a plane parallel to that of xz at the distance y. Similarly, z^ and z^^ or q and t are the differen- tials of z with respect to y, x being constant ; that is, z may now be regarded as the ordinate of the section cut by a plane at right angles to the first, and at the distance x from the plane of yz. Therefore, as x and y receive no variation, we we must have, as heretofore, dz' or Sp — — — , 8z" or Sr — —^-r-, ax ax dSz . .^ d^Sz dz^ or dq = ——, oz., or 6t = 2 ' dy ' '' dy and these equations, which are exact, may be used in any manner we find convenient in transforming SU. 363, Considering for the present the terms of the first order only, we have <^^=/"X"5 F,fo + rp(J/+ V,SqUfydx = o. (5) But without entering upon any general discussion of the con- ditions which must hold in order that (5) may be satisfied, let us return to our original problem. Here V=Vi+f + ^\ V, = o, K= J V - ^ SURFACE OF MINIMUM AREA. 43 1 SO that (5) gives Now we cannot assert that every element of this integral must vanish, because we have also required that the edges of the surface shall be fixed — that is, that dz, for all points of the edge or contour, shall vanish — and this condition has not yet been imposed upon dU. Indeed, there is an analogy between the present problem and Prob. I. For in Prob. I. we were to connect two fixed points by a line of minimum length, requir- ing us to minimize a single integral ; while in the present prob- lem we are to connect an infinite number of fixed points, forming the given contour, by a surface of minimum area, re- quiring thereby the minimizing of a double integral. 364-. The condition just mentioned may be imposed some- what as in Prob. I. For we have dqdy dx ^0 ^'Vo |/i_|_^2_|_^2 Szdx '^0 lyo ^j -f/-(-^^ If now we remember that for any abscissa x, y^ and y^ are the two ordinates of the projected contour corresponding to this abscissa, we shall see that the ^'s corresponding to y^ and y^ re- late to the edge or contour only of the required surface, and that therefore every dz in the single integral in (7) must van- ish, causing the integral itself to vanish. Now since we may adopt either order of integration in a 432 CALCULUS OF VARIATIONS. double definite integral without affecting its value, we may write _ P_ ^^ dy _ p l>y,d p_ ^^ ^^_ Here we regard y as the independent variable in the equation of the projected contour^ so that x^ and x^ are always ordinates of this contour, y being the abscissa. Hence, as before, every Sz in the single integral of the last equation refers to some portion of the contour only, and must vanish. Hence, finally, we must have 365. It is here necessary to notice two points. First. It will be seen that the form in which the terms under the sign of single integration— which terms are the terms at the Hmits in this problem— have been left is incongru- ous, inasmuch as we do not retain the same independent vari- able throughout. But our only object at present is to show that when the contour is fixed the terms at the limits will vanish. Indeed, the arrangement of these terms in the case of multiple integrals, so as to enable us to discuss with any- thing like generality the conditions which must hold at the SURFACE OF MINIMUM AREA. 433 limits, has proved to mathematicians one of the most difficult points connected with the calculus of variations. For although this subject had more or less occupied the attention of Gauss, Poisson, Ostrogradsky, Jacobi and Delaunay, the last of whom has been followed by Prof. Jellett, it remained for M. Sarrus to present a method of treatment which has the merit of being systematic and general, and is perhaps as nearly f»er- fect as the nature of the subject will permit. Second. The two differentials in (9) denote the entire change produced in the first fraction when we change x into X + dx, and in the second when we change y into y -\- dy, p and q being variable both for changes in x and y, so that, with respect to x or y only, these differentials may be said to be total. Such differentials are, however, called partial, since they denote the change incident upon an alteration in one in- dependent variable only, while there are two which might be varied. 366. Now the two single integrals in (7) and (8), taken to- gether, certainly involve every dz for the contour, which would not perhaps be true of the first integral alone if a por- tion of the projected contour should be a right line perpen- dicular to the axis of x, nor of the second if a portion of the projected contour should be a right line perpendicular to the axis of J. Hence in (9) the condition that Sz shall be zero throughout the entire contour has been imposed. Now as the sign of Sz for every point of the required sur- face is wholly within our power, and its value is subject to no other restrictions than that it shall be infinitesimal, and shall render dp and Sq also infinitesimal, it will appear, as hitherto, that we can only satisfy (9) by equating M to zero, so that we shall have A^--^^-^-^±-—i-_-^^-M=o. (10) dx Vi-\-p'-\-q' dy 4/1+/ + ?" 434 CALCULUS OF VARIATIONS. Performing the differentiation indicated in the last equation, observing that p^ — q' — s, we have, after reducing to a com- mon denominator, This expression, which is a partial differential equation of the second order, is known to indicate that the required sur- face must be of such a nature that at every point the principal radii of curvature may be equal and taken with a contrary sign, so that their sum may be always zero. Moreover, equa- tion (lo), which is the fundamental equation, would evidently be satisfied by a plane, since / and q would then become con- stant. This could not, however, as we have already shown, be the general solution, because, if the given contour were not a plane figure it would not be possible to make a plane sur- face fulfil all the conditions at the limits ; that is, to pass through every point of the given contour. But we shall resume the consideration of (ii) presently. 367. Assuming the required surface in any particular case to have been determined, let us now examine the sign of the terms of the second order. Since c does not enter V expli- citly, we have, from (2) and (4), 5 = Vpp Ji> ^0 ai^d jTj. For we must remember that this contour encloses a certain plane surface, and that the general values of X and y, as used in the double integral, must include the co-ordinates of every point of this plane surface ; so that for every value of x there are an infinity of values for 7. If, there- PROBLEMS LVII. AND LVIII. RESUMED. 455 fore, we regard x and y as varying throughout the integral, as we evidently may, we in effect suppose the points of this plane to change their positions. But this is manifestly use- less, it being sufficient to add to the ordinates }\ and )\^ and to the abscissas x\ and x^, infinitesimal increments Dy^ Dy^, Dx^ and Dx^, these quantities being independent of either sign, and representing the same increments as would be denoted by Sy^ and dx^ and dx^, if we had been obliged to vary in the same manner the form of the projected contour in the case of a single integral, the limits being also variable. 394. Let us now consider in detail the mode of obtaining SUy where U =J^ J Vdydx, V being any function of x, y, z, p and q, the limiting values of x and y being also subject to variation. First, varying z^p and q only, we have, to the second order. r^ r\ V,6z+ VpSp+ V^Sqldydx + 2V^qSpdq + Vq^Sq^\dydx. (l) This gives to the second order the change which 6^ will under- go when we pass from any primitive to any derived surface, the form of the projected contour or of the bounding walls remaining unaltered. In the second place, let us consider the change which [/ will undergo when we alter the dimensions of this derived surface in any infinitesimal manner we please, supposing x^ and jTj to remain unchanged. Since U consists of the sum of the elements dx I Vdy, in which the integration is entirely independent of x^ that quantity being regarded merely as a 45^ CALCULUS OF VARIATLONS. constant, if it enter Fat all, the change sought will evidently be the sum of the additional changes which each element will undergo if, after having varied z, p and q only, we also vary /o and jKi by adding any increment or decrement, Dy^ and Dy^, Proceeding, then, with one of these elements as if it were the only integral in question, we shall obtain, in addition to the terms arising from the variation of ^, / and q only, which are already included in (i), the terms C 1 ^'^y + 1 ^' ^y^ +^vDy\, (2) where and K^Vy^V,q+V^s+V^t (3) 6V= V,Sz+ Vpdpi-V^Sq, (4) Hence, summing the changes in all the elements, we have VDy-^r\v,D/ + 6VDy \ dx, (5) which gives the change sought, and must be added to (i). 395. We must now consider, in the third place, what change U will undergo when we make any infinitesimal changes in the values of x^ and ;r,. It is easy to see that we can, if we choose, pass from the primitive to the derived surface by first making the necessary changes in y^ and y^, or in the form of the upper or lower por- tions of the projected contour^ x^ and x^ remaining fixed, as also the form of the surface ; and then varying the surface under the supposition that z, p and q vary, and also that x^ and x^ become respectively x^ -f Dx^ and x^ -\- Dx,, the new limit- ing values of y, which are y^ -\- Dy^ and y^ -\- Dy^, remaining fixed. Now the portion of c^C/ which will arise from varying PROBLEMS LVII. AND LVIII. RESUMED, 457 this surface, supposing x^ and x^ to remain fixed, is, as we have seen, found by taking the sum of equations (i) and (5), so that we have now to determine the portion which will arise merely from changing x^ and x^ into x^ + Dx^ and x^ + Dx^, and this added to (i) and (5) vvill evidently give the complete variation of U to the second order. 396. Now by the changes in the limiting values of j alone, U will become P^\ f*V\-\-'Dyx nx-i I ^ Vdy dx or / vdx. (6) where V =fjydy^ly VDy + etc.\= v^+l^ VBy + etc.}. (7) Now since v does not contain the limiting values of x, either explicitly or implicitly, any element vdx will be independent of any changes in these limiting values, and therefore, although 2^ is a definite integral, we may employ the same reasoning as though it were not, and say that the change in J^ vdx, due to the variation of the limits x^ and x^, must be 1: 1 ■' Dx-\^- v' Dx" + dvDx \ . (8) Let us now approximate in (8) as far as the terms of the seconS order. We have = /"' f' VdyDx^ r- l^'VDyDx, (9) where we mu^t remember that the last term represents four terms involving merely the values of V, Dy and Dx at what 458 CALCULUS OF VARLATIONS. we may term the four corners of the surface, although Dx^ and Dx^ are infinitesimal and independent constants. Now in reducing the second and third terms in (8) it will evidently be sufficient to regard v as merely equal to v"^. Then, by equation (i), Art. 375, we have and therefore, to the second order, i^^\ dv /^i I py^dV /^i /y^i dv „ , X / l.'^D^-^^ 1/ ^dyDx' + l I ~Vi-Dx\ (10) ^^0 2 dx '^0 2 ^^0 dx -^ ' '-^0 '^0 2 dx ^=V'=V,+ V,p + Vpr + Vqs, (II) where accents as usual denote total differentials. We shall have also, to the second order, r^dvDx = I'^'S r^VdyDx = r^ r'dVdyDx, (12) where SV has the value given in (4). Hence, adding equa- tions (9), (10) and (12), we obtain, for the last portion of ^U, '^C-2 V%Dx'-^XyVdyDx } . ■ (13) 397, Now by adding (i), (5) and (13), and then substitut- ing the values of V, V^ and ^Ffrom (3), (4) and (11), we shall have the complete variation of U to the second order ; and if U is to be a maximum or a minimum, the terms of the first order must vanish, while those of the second must become in- variably negative for a maximum and positive for a minimum. PROBLEMS LVII. AND LVIIL RESUMED. 459 As will be naturally surmised from their complicated na- ture, the determination of the sign of the terms of the second order transcends our present knowledge of variations, even when the form of V is known ; and we shall therefore in future consider only those terms in dt/ which are of the first order. Collecting these terms, we have +X'X' 5 ^'^^ + ^^^^ + V^6q\dydx = O. (14) iVow transforming the double integral as in equation (4) of the preceding problem, we shall have, finally, + r^ r^ VDy dx-\- T' T" VDx dy 398. Now it will appear, as in the preceding problem, that because the part of (^ Sunder the sign of double integra- tion cannot depend upon terms which relate to the limits only, these two parts must be independent, and that L and M must severally vanish. Therefore we see that here, as in single integrals, the differential equation from which the general solution must be obtained will be the same whatever may be the particular conditions which may be imposed at the limits. Let us then examine the equation L —o. It is easy to see 4^0 CALCULUS OF VARIATIONS, that if we can regard the quantities dz^ Dy and Dx at all the limits as independent, the four terms in L will be also inde- pendent, and we shall be obliged to equate them severally to zero. Hence, using k as in the last problem, we must have / / ^kdzdx = o, / / 'Vridsdy = o, \ (i6) , .- ri'-VDydx = o, r- r-VDxdy = o, t/xo /yo Ixq Ugo -^ J in which the first two equations give the same conditions as in that problem. Now in the third equation we must remember that Dy is perfectly in our power for every point of the upper and lower portion of the projected contour, and is in fact what might be termed ^y^ if we had not agreed to suppose x and y incapable of receiving any variation ; so that this integral will not cer- tainly vanish unless we have/ F= c In treating the fourth equation, we must remember that Dx^ and Dx^ do not, hke Dy^ and Dy^, denote an infinity of quantities, but signify only one each, so that they are each arbitrary constants, and we must have/ / Vdy — o, and we cannot make any further reduction, because the integral is definite, and none of the quantities involved are in our power, F= o. We must then, in the present case, have the equations k^o, V=o, Fp = o, £'Vdy = o; (i?) the first two equations holding along the upper and lower con- tour, and the last two along the ri^/it and left. Or, as in the preceding problem, the condition Vqdx — Vpdy = o must hold for the entire contour ; while we now add that the condition PROBLEMS LVII. AND LVIIL RESUMED. 46 1 F = o along the entire contour will satisfy all the remaining requirements of the limits, and will be necessary for all but the right and left portions of the contour, which might, per- haps, be satisfied by some other condition also. 399. But as it is necessary in the case of curves to im- pose some manner of restriction upon the extremities in order that ^may become a maximum or a minimum, so in the present case it is easy to see that the required surface cannot possess a maximum or a minimum property unless its contour be subjected to some sort of restriction. Now the most general case which will arise is that of our problem — namely, where the required surface is to have its contour upon t)ne or more given surfaces — and this case we will now proceed to consider. 4-0 0« Let the equation of any one of the limiting surfaces be of the form dZ = PdX^QdY or Z = /(X,F), (18) and let us first suppose it to be touched by a portion of the upper contour. Now if we pass a plane parallel to that of yz, at any distance x from that plane, the sections cut from the required and the limiting surface will be two plane curves, which meet, and the equation of the curve cut from the limiting surface is dZ = QdY, while that of the other is ds = qdy. Therefore, so far as these two curves are con- cerned, we may regard y as the independent variable, and x as a constant, if it appear at all in their equations. Hence, when we change y^ into y^-\- Dy^, we may employ precisely the same reasoning as in Art. 69 ; so that, since Q would replace/"' in that article, we shall, neglecting terms of the second order, have, as in equations (2), Art. ^6, Sz,= {Q-q),Dy,, and a similar equation will hold for the lower limit. 462 CALCULUS OF VARIATLONS. In like manner, for the limiting surface at the right, by passing a plane parallel to that of xz at any distance y from that plane, we find and a similar equation for the lower limit. Or to render these equations more intelligible, we may write or, to the second order, we shall have # ly. =iy^i^Q-g)Dy+'-{T- t) Df- SqDy\,^ y (20) £"& =£' I {P-p)Dx\\^{R-r)Dx'-SpDx 401. Now since equations (19) restrict the independence of ^z and Dy^ and dz and Dx at both limits of y and x, equa- tions (17) will no longer hold true. But from (15) we may write ^ =£?ly''^ VDy)dx + l^jj;;\V^6. + VDx)dy = o;{2,) and eliminating Sz by (19), we have + rX'' ^+ Vp{P-p)]Dxdy = o. (22) Now it is evident that the quantities By^, By^, Dx^ and Dx^ are entirely independent of one another, as the fact that the con- tour is to be confined to certain surfaces in no way restricts PROBLEMS LVIL AND LVIIL RESUMED. 4^3 US in varying the form of the projected contour. Moreover, as before, Dy is completely in our power for every point of the upper and lower contour, while, for either limit of x, Dx is an arbitrary constant. Therefore, by the same reasoning as in the former case, (22) must give the equations V^k{Q-q) = 0, £'\V+V^{P-p)]dy^O; (23) the first holding along either the idpper or lozver portion of the contour, and the second along either the right or left. , But >^ = Vq — Vpj/x ; and also tl2 = pdx + qdy and dZ = PdX + QdY, the first being the equation of the required, and the second of any limiting surface ; and since along their intersection ;r, j/, ^ and X, V, Z are identical, we must have along such intersec- tion dy _ P-p pdx -f- qdy = Pdx -\- Qdy, dx Q — q Substituting this value in k, and then the result in the first of equations (23), the conditions at the hmits finally become V+Vp{P-p)+Vq{Q-q) = o,-] r'\V+V,{P-p)\dy = c (24) To discuss the terms of the second order we must employ equations (20) in the place of (19), proceeding as before, and setting aside all terms of that order which may arise. Then we shall have the same terms of the first order as before, while those which we have set aside must be added to the terms of the second order which we have already exhibited in equations (i), (5) and (13), thus rendering the complete 4^4 CALCULUS OF VARIATLONS. terms of that order still more complicated, and the deter- mination of their sign a much more hopeless problem than before. 402. Now the first of equations (24) must hold along the entire upper and lower contour, and may represent as many distinct conditions as there are limiting surfaces touched by these portions of the contour. The second of these equations holds along the right and left contour only, and will be satis- fied if we suppose the first to hold for these portions of the contour also, because along these portions, being parallel to the plane of yz, q and Q are equal, so that Q — q will vanish. The first condition, then, is necessary for the upper and lower, and will satisfy the requirements of the limits, should it hold throughout the entire contour, although the right and left portions may furnish some additional condition. 4-03. Let us now apply the foregoing theory to Probs. LVII. and LVIIL, beginning with the former. Here it is easy to see that equations (24) give the condi- tions ' + ^ + ^^ = °' ^(i^i^^^-- (^s) The first equation denotes that the required surface must meet at right angles all the limiting surfaces which are touched by its upper and lower contour, and the same condi- tion might also prevail along the right and left portions, although we cannot assert that the second of equations (25) might not be satisfied in some other manner. In general, however, the projected contour will be a closed curve, in which case the right and left portion reduce to points, causing the second equation to disappear, and the first to hold along the entire contour. As before, if we could obtain the general integral of equa- tion (10), Prob. LVIL, which would involve a number of arbi- PROBLEMS LVII. AND LVIII. RESUMED. 465 trary functions, not exceeding two, it would be necessary to determine these functions in such a manner as to satisfy equa- tions (25). 404-. Let us now turn to Prob. LV^III. Here equations (24) become V — mx{P—p) — my{Q — q)\ = o^ H'v^n - 1 1 7; — mx{P — p)\dy — Q. (26) These equations will both be satisfied hy v = o throughout the entire contour, which supposition would, as before, lead necessarily to a conic surface. Neglecting this supposition, we have V — mx{P — p) — iny{Q — q) — o, qj = 2 — px — qy. Whence, substituting and transposing, we have - m{Ppx + Pqy) + {m — i) {px-^qj) ^ — z. Adding mz to both members and transposing, we have m{z — Ppx — Pqy^ = (m — 1) {z — px — qy). Whence z — px — qy ni z — Ppx — Pqy fn (27) which shows that if at any point of the upper or lower con- tour tangent planes be drawn, the first to the required and the second to the limiting surface, the portions of the axis of z comprised between the origin and these planes respectively will be to each other ?iS mis to m — i. 406. Having now reached the general discussion of the problem, let us consider more particularly the mode of deter- 466 CALCULUS OF VARIATIONS. mining the arbitrary functions in the various cases which may arise. First suppose the contour to be a fixed boundary, and let it, for example, be a circle of radius a, having its centre on the axis of z, and its plane parallel to that of xy at the distance c. Write — = / and n = — - — . Then, from the equation of X I — m the contour and from the general equation of the surface, which now becomes z = x^F{t) + x/Xt) = x^ F+ xf, (28) we have / {^+y) = ^\ / =^- = I, / ^ = c, ' ' a * (29) Having solved the last equation for/', we may then omit all signs of substitution, because the form of /' must remain the same for all values of x and y belonging to the required sur- face. Hence we have /r+? ( ^ _ a:^F \ ^~ a \ i/(T+7^)^(* (30) Now restoring the value of /, and substituting for/' in (28), we obtain As the lower limiting values of y furnish the same equations as the upper, we have no other condition by which to deter- mine F, which may therefore be assumed arbitrarily. PROBLEMS LVII. AND LVIII. RESUMED. 467 Next suppose two circular arcs situated as before, having radii a and a' , and that the given contour is to consist of a portion of the upper arc of each circle joined by any two curves whose projection^ on the plane of xy shall be the right lines x — x^ and x — x^. Then y.^ and y^ belong to these arcs only, and we obtain, as before, =/ 4/(1+0 V(l+/=)» ' VI +1' (32) Solving these equations for/' and F, and omiting the signs of substitution for the sanle reason as before, we have Substituting these values in (28), we obtain z(aa'^ - a'a^) = Qa'^ - ca^) Vx' + /+ {ca ~ ca') V(?+7)'^.(34) Thus we see that the two functions will be determined by the circumstance that the required surface is to pass through the two arcs, and we cannot impose any further conditions. Unless, therefore, the remaining portions of the fixed bound- ary be so assigned that they would lie necessarily upon this surface, the conditions of the problem cannot be all satisfied. We shall, however, have occasion to consider these functions again presently. 406, Let us next suppose that the required surface is to connect two planes whose equations are 2 = ax -\- dy -{- c and ^ = a'x + d'y -|- ^'. (35) 468 CALCULUS OF VARLATLONS. From equation (5), Art. 372, observing that m±2_^ and n = -^~, (36) we have v, or the numerator of (27), equals fn — I ^F, (37) while the denominator of that equation must become either c or c' . Hence (27) furnishes the conditions /Vx IVo {m + 2)x'^F = mc, / (^ + 2)x'^F = mc', (38) We have also, along the upper contour, z — {a-[- bt)x A-c, z = x^F-\- xf\ (39) Eliminating z and x^F between these and the first of (38), we obtain 2c '' = (m-^2){f -a~bt) '* (4^^ and substituting this value in the first of equations (38), we have {f - a — btf = — {m-\- 2y-'^c''-^ F\ (41) and in like manner we find, along the lower contour, {f -a' - b'ty^ = — (//^ 4- 2)1 -nc'n-\F, (42) fi I If we solve (41) and (42) for/' and F, and put / for , we iz shall obtain PROBLEMS LVII. AND LVIII. RESUMED. 4^9 t — ^/p _ ^p ' \ (43) , , s . \a-\-bt — a' — b't\'^ \ F=m(m + 2Y- \ \^,,_^,^ f • J Although in (4 1) and (42) f belonged respectively to the upper and lower projected contour only, in (43), for the rea- son already explained, it may belong to any point whatever of the required surface. Hence equation (28) becomes, after re- storing the value of /, _ c'^iax -^ by) — cP(a' x + b'y) ^ ~" c'P — c^ , ^ ^ \ax-\-by — a'x — b'y}'^ . ^ + m{m + 2)^-1 I ^^^-^.^ _ ^^^ -^ [. (44) 407. It will be remembered that, in the case of maximiz- ing or minimizing any single integral U, it is necessary, in order to render the method of variations applicable, that no element of U ox oi STJ shall become infinite within the range of the integration ; and it will readily appear that when ^is a definite double integral the same principles will apply, since each element of U is treated precisely as before. Now from (37) we have, in the present case, U=r'r\'^ dydx — fj'f^' c^^^nrnpyn dy dx, (45) fyi I 2 "^fn where C = ■ — . But nm = ; so that it will appear, m — \ \ — m ^^ upon a little reflection, that nm must be negative except when m lies between zero and unity. Hence when x ^ o, x'^'^ must become infinite ; and it will appear that to prevent v from be- coming certainly infinite, or at least indeterminate, we shall 470 CALCULUS OF VARIATIONS, be obliged to make F vanish throughout U^ which will give z — xf\ thus bringing back the conic surface, in which we must, as V is zero when x is zero, still reject all values of m which would render v^, ^m-i qj. ^m-2 infinite, since the second and third of these quantities occur respectively as factors of the terms of the first and second orders. In this case, there being but one function to determine, the first supposition in Art. 405 would determine the surface completely, requiring a right cone ; so that In Art. 392 f would, as we have seen, remain indeter- minate, and indeed it is easy to see that we could have no finite minimum while the limiting values of z remain variable. In this case equation (27) is inapplicable, since in obtaining it we assumed that v did not vanish. Problem LXI. 4-08. It is required to determifie the form of the surface zvhich will maximize or minimize the expression u--£?fy!' ^/+? ^yd- =£?£' vdy<^'- (I) Here and observing that/, — / =: s, the equation M = o will reduce to q'r -2pqs+p''t = 0. (3) This equation may be integrated by the method of Monge ; ADDITIONAL PROBLEMS. 4/1 and adopting the notation of De Morgan, page 719, we have, putting z for u, R = q\ ^=-2/^, T=p\ a = q^df-{- 2pq dy dx -\- p^dx^, V — o, P G = q^dp dy + p'dq dx, -^ — — ^, dz ■= pdx -f qdy. Now if a vanish, we see from the last equation that dz will vanish also, and vice versa; and by theory cr will also vanish. Substituting j.i dx for dy., the equation o" = o gives qdp — pdq = o. i> Whence we may write - = —f{z) — —f. Again, when or = o, we have dy — f^dx or dy-\-^dx or dy — fdx-=o, (4) where we must remember that / is to be regarded as a con- stant, because dz is zero or ^ is a constant. Hence y—xf= F{z) = F. The complete integral of (3) is, then, y = xA^) + F{^) = ^f+P, (5) where / and F are any functions of z whatever. 409. Let us first suppose the limiting values of x^ya-ud z to be fixed, or that the surface is to pass through some fixed boundary, and let us require, as a particular case, that two portions of this boundary shall be given by the equations /v+/)=^s r^=r^, (6) /V+/)==^'^ /%./--$. (7) 472 CALCULUS OF VARIATIONS. Then, for the upper limit, we have zx _ a a am ^,^1.^ i/:+f; ^-" + ^ m r I + =^, r I + ^ m az y = Vm'-\-z' Therefore, by (5), we have, for the upper Umit, az __ ^^^ _ I p^ (g^^ ^/W^z'' Vm' + z' Similarly we obtain, for the lower limit of j, a'z _ a'm'f Vm''' + z' Vm' + F, (9) Solving for / and F, remembering that the results will no longer refer to the contour only, but will hold for every point of the required surface, we shall obtain a Vm''' -{-z" - a' Vnf+ f=z am ym'^ -\- z^ — a'm! Vm^ -\- ^ „ _ aa\m — m-'^z am Vjn'^ -\- z^ — a'm' Vm^ -\- ^ Now if the surface determined by the substitution of these values of /and i^in (5) do not necessarily fulfil all the require- ments of the problem regarding other portions of the fixed boundary, we conclude that these conditions cannot all be satisfied. 410. Next, suppose we give merely the limiting values of X and y^ those of z remaining variable ; that is, suppose we ADDITIONAL PROBLEMS. 473 give merely the form of \h^ projected contour, or of the cyHn- drical walls. Then we see from (2) that equation (12), Art. 390, will furnish the condition qdx — pdy — o ; (lo) which shows that the required surface must meet these walls at right angles. To discuss the form of the functions, let us suppose the wall to be a right circular cylinder, having the axis of z as its xdx axis. Then along the projected contour we have dy ^ , y and (10) gives, by substitution, px-\-qy = o. (II) But by differentiating (5) with regard to x and y respectively, we find — xf y xfA-F px = - / , qy = . , - = ;; ' . Hence (11) gives F^o; and (5) becomes y = xf, which may evidently be put under the form The function /~* will remain undetermined unless we as- sume some other form for a portion of the cylindrical wall. Suppose, then, another portion to be elliptical, giving bx dx -^ ay Then along it we have, as before, # bpx -\- ^qy = O' (13) hold > px-. ^xff -v,=.- -ft Whc ;nce (13) gives (fl -b)ff -I 0, 474 CALCULUS OF VARIATIONS. y Putting / for -, we shall have from (12), which raust now qy'=^yft "ty=f, »^. 411. Let us next suppose the edges of the surface are re- quired merely to rest upon one or more given surfaces. Then, substituting from (2) in equations (24), Art. 401, we find the conditions at the limits to be Pp-\.Qq^O, r'-^=^M=^dy=:0. (14) The reader can readily apply in any particular case the first of these conditions to the determination of the arbitrary func- tions. 412. When the limiting values of x and y are fixed, whether those of z be subject to variation or not, we find the terms of the second order to be Hence, since we suppose the denominator of (15) to be posi- tive, we may conclude that U will become a minimum for ali solutions which do not give rise to infinite values for any element of dU] unless, indeed, it be possible to assign such values to Sp and Sq as will cause every element of (15) to vanish. ADDITIONAL PROBLEMS. 475 Problem LXIL 413. It is required to determine the form of the surface "juhich will maximize or minimize the expression / / \z — px — gy) dy dx = I ^vdydx, (i) while at the same time the vacations of p and q are always to be so taken that the expression may always have an assigned constant value. This is evidently a problem of relative maxima and mini- ma, and we can treat it by Euler's method precisely as in the case of single integrals. For, supposing first the limiting val- ues of x, y and z to be fixed, the reasoning of Bertrand, ex- plained in Art. 93, which the reader is supposed to re-peruse, can, in the following manner, be extended to this problem. Since the terms at the limits vanish, we must have / / 'Svdydx or / / ^VSzdydx = o. e/Xo e/Z/o *Ixo <^2/o r^ r^Sv'dy dx or f' H" V'dzdydx^ o ; (3) 'Xo t/yo vxo t.fyo where 77 — 7, _ ^!i _ fl^ V — '?) ' — '^^'^ — ^- (a\ "^-^^ dx d/ ^ ~^" dx dy ' ^4^ Now suppose the required surface to have been obtamed, and on it select any two portions in such a manner that for every point of either portion, when that portion is considered sepa- rately, both V and V may preserve an invariable sign. Then 4/6 CALCULUS OF VARIATIONS. vary z throughout these portions only, leaving the remainder of the surface unvaried in form. Also make the sign of dz invariable throughout each, giving to it in the two portions like signs when those of V are unhke, and vice versa. In this way, by giving suitable values to Sz, we can, as in Art. 93, satisfy the first of equations (3). But the second of these equations may be written X X '^'^y''^ = X. Jy, fVS.dydx, /=f ; (5) the variations of z being taken as before ; so that unless f be a constant, we can certainly effect that the double integrals taken throughout the two portions shall be numerically un- equal, and hence the second of equations (3) would not be satisfied. The remaining reasoning, by which the necessity of Euler's method is established, is precisely like that of Art. 93. If the limiting values of x, y and z are also subject to varia- tion, the method of Euler is still equally applicable. For sup- pose the required surface were to be bounded by certain cylindrical walls or by certain surfaces. Then, since we are not compelled to vary the limiting values of x, y or ^, the re- quired surface must evidently be of that kind which will sat- isfy all the conditions of the problem when the contour is to be fixed, the only question being to determine the conditions which must hold along the contour ; and since, in double as in single integrals, the fundamental equation obtained in discuss- ing any problem of absolute maxima or minima is the same whatever be the conditions which are to hold at the limits, the appHcability of Euler's method is apparent, as in Art. 96. 4(4. We see, then, that we are in the present case to dis- cuss the conditions which will maximize or minimize abso- lutely the expression ADDITIONAL PROBLEMS. 477 ^ = £?J2'^^ -P^-qy^a •//+?) dy dx Here F,= i, Fp=-;r + — ^=, F^ ==__;/ + _^£= ; (2) SO that, writing ^ z= — — , the equation M = o gi^^es ^V - 2J>qs +/V = ^(/' + /)». (3) This equation is integrable by the method of Monge. See Boole's Diff. Eqs, Chapter XV., or De Morgan's Diff. and Integ. Calc, page 719. Adopting the notation of the latter, we may write Zp_ ^' C_ 2/^ 7-_ / jr_r yp -\-q)^ ^ _ q\dq dy - dpdx) + 2pq dqdx , ^ ^^ (7+?? + ' dz ^= pdx -\-qdy. Now the condition a: = o renders dz zero, and also gives dy — }xdx =:. Q\ so that we may write ft dx = - ^+/.(^) ^-y +/.. (A) 478 CALCULUS OF VARIATIONS. Now substituting pidx for dy m. g and r, which must also be- come zero when a is zero, we obtain Integrating these equations, we obtain _Z__ z = -b r^dx^ flz\ --=i= = -bx +fAz\ (B) Now by squaring and adding equations (B), and substituting from (A) the value of / - dx, we shall obtain the integral sought. The complete integral of this equation is, therefore, L = (^ +/(^))' + (j/ + F{z)y = {x +/)' + (j. + F)\ (4) 415. This equation is easily interpreted. For suppose a circle whose radius is — ; and while keeping its plane always parallel to that of xz^ let its centre move along some curve in space whose equations shall be X^-f. Y=-F, Z^z, (5) Then it will readily appear that (4) represents the equation of the surface generated by the circumference of this circle as it moves along the given curve, and that when we shall in any particular case have determined the form of the two arbitrary functions, / and F, we shall know the nature of the curved directrix of this surface. When the contour of the surface is fixed, the functions must be determined in accordance with this condition. ADDITIONAL PROBLEMS, 479 If the bounding walls only are to have a given form, equa- tion (12), Art. 390, will give (/ ^P" + / - (^Q¥x = {x y/ -^q" - ap)dy, (6) and / and F must be determined so as to satisfy this equation. When the required surface is to be limited by one or more given surfaces, the first of equations (24), Art. 401, which is the only one of importance, will become, by substituting from (2), ^/ + q" ^^^ and / and F must then be determined in accordance with this condition. 416. Of these cases we will consider but one — that in which the required surface is to be limited by two planes, each passing through the origin, and having for their equa- tions z ^ ex -\- c^y, 2^=- c'x -\- c^y, (8) In this case (7) will give l''\cp + c,q) = O, /\j> + clq) = a (9) But from (4) we obtain /=- -+^ ? = - (10) (^+/y.+0'+^)^/j Hence, by the use of (8), equation (9) gives /'\^J^cf+c,F)^o, /%J^c'f+c,'F)=o. (II) 48o CALCULUS OF VARIATIONS, From (5) these conditions may be written Z=cX-^c,Y, Z=c'X+c/Y. (12) From these equations it at once appears that if there were but one limiting plane, the centre of the generating circle would be compelled to remain always in that plane, and that in the present case the centre must move along the intersection of the two limiting planes. This will give an oblique cylinder having a circular base, the line in which the two planes inter- sect being its axis. We can, of course, determine / and F in the usual way, thus obtaining, from (11), cc^ — c c^ cc^ — c c^ Moreover, when in any particular case we have determined the functions / and F^ we shall then be able to determine also the constant ^ or — — . For, as in the case of single integrals, we have the condition that one of the double integrals is to re- main constant, and we may suppose a definite value to have been assigned to it. 4 17. In considering the terms of the second order the same reasoning will hold as in the case of single integrals. For the variations of ^,/ and q are subject to a certain restric- tion which we cannot explicitly express, and the method of Euler will cause the terms of the first order to vanish whether these variations are restricted or not. But the variations are still restricted, and when we come to the terms of the second order it is conceivable that even when they do not indicate a maximum or a minimum, the variations being unrestricted, they would do so if we could employ such variations only as would permit one of the double integrals to remain always constant, which, however, we have no means of doing. But when these terms indicate an absolute maximum or minimum SURFACE OF LOWEST CENTRE OF GRAVITY. 48 1 — that is, for all systems of variations — there would seem to be no doubt as to the existence of a relative maximum or mini- mum also. In the present problem, when the limiting values of x and y are fixed, the terms of the second order are the same as in equation (15), Art. 412, only multiplied by a. Hence we may in this case conclude that t/will be a maximum or a minimum according as a is negative or positive. Problem LXIII. 4- (8. It is required to determine the form which a surface of given area whose edges are in some manner confined must assume in order that the depth of its centre of gravity may be a maximum. The given area is and assuming the axis of z vertically downward, we have, for the depth of the centre of gravity, which is to be a maximum. Or, since A is to be a constant, we may say that / ' f^^z V i -\- p"" -\- q" dy dx is to be a maximum, while J^ J 4/1 +/' -\-q^ dydx is to remain constant. Hence, employing Euler's method, we may write ^=X?fyy - ^) ^^i+f + fdydx =£;£vdydx. (I) 482 CALCULUS OF VARLATIONS. Here F^r.: |/i -[-/ + ^^ Y ^ (^ - d)p p ^ {z-d)q^ (2) Hence the equation M = o reduces to i+/ + ^^-(^-^^)Ki+/)^-2/^^ + (i+/)^}. (3) This equation is not integrable ; but calling R and R ' the principal radii of curvature, and estimating the signs properly, (3) may be written because =r — : — - y (4) R R' {z-a)Vl-\-f + q' Equation (4) shows that the mean radius of curvature of the required surface at any point is twice the normal ex- tended until it meets the plane whose equation is ^ = <^. The same equation also indicates an analogy between this surface and the catenary, which gives, as we have already seen, the solution for a similar problem relative to plane curves. (See Art 282.) If the contour, instead of passing through some fixed curve, be confined to certain cylindrical walls only, we must have, from equation (12), Art. 390, qdx—pdy^=^o, showing that the surface sought must meet these walls at right angles. When the edges of the required surface must be upon one or more given surfaces, the equation of any one of which is dZ = PdX -\- QdY, the first of equations (24), Art. 401, will give the condition i -{- Pp -\- Qq = o, showing that the required surface must be normal to the limiting surfaces. COVERING SURFACE OF MINIMUM AREA. 4^3 Problem LXIV. 4- 19. It is required to detennirie the form of the surface whose area shall be a minimum, and which shall cover a given volume on a horizontal plane. Here, since the given volume is / / zdydx, we may write at once Here and the equation M = o will give (I + q y- 2pqs+{l +f)t ,l_ ^ ^ V^T+f + T? ^ * ^^^ This equation, which is not integrable, gives, as in the preced- ing problem, by a contrary estimation of signs, R + J' = a- (^ Hence the required surface must be such that its mean cur- vature at every point may be constant. 4-20. We already know that it will be necessary to the existence of a maximum or a minimum that the contour shall either be fixed or rest upon some surface or surfaces, the cal- culus of variations affording in the first case no further equa- tions; and we are unable to integrate (3). But when, in the 4^4 CALCULUS OF VARIATIONS. second case, these limiting surfaces are certain cylindrical walls normal to the plane of xy, equation (12), Art. 390, gives qdx—pdyz=zo^ (5) the meaning of which we know. When, however, the limiting surfaces to which the contour is to be confined may have any given form, the first of equa- tions (24), Art. 401, gives ^ Vi +/ + ^^ - a(i +Pp+ Qq) = o. (6) Suppose, for example, the limiting surface to be a plane whose equation is ^ = ^ Then (6) will give (7) Hence the angle A which the tangent plane to the required sur- face at any point of the contour makes with the plane of xy must be a constant, since the first member of (7) is or sec A cos A. When h — o, we must have cos ^ = o, and the required surface meets the plane of xy, and is normal to it. The sur- face of a hemisphere of radius 2a would evidently, in this case, satisfy all the conditions of the question so far as the terms of the first order are concerned ; but a satisfactory investigation of those of the second order would probably be impossible. When the limiting values of x and y are fixed, the terms of the second order may be written and, as in the case of a spherical surface, the radius is 2a, and is positive, we may conclude that if we vary the form of the surface only, the circular base remaining unvaried, the surface will be a minimum. EXTENSION OF SARRUS'S METHOD. 485 421. To give a more comprehensive view of the method of M. Sarrus in the treatment of double integrals, we now pro- ceed to a more general problem. But the reader who desires may omit the discussion of the following example. Problem LXV. // is required to maximize or minimize the expression U:= I I Vdy dx, where V is any function of x, y, z, p, q, r, s, and t. It is evident that, supposing the limiting values oi x and j to be also variable, we shall have dU= r^ r^VDydx+T" r^VDxdy + Vr^r-^ Vs<^s + Vt^t]dydx = o, (i) Now all the terms except the last three are to be trans- formed and arranged as in equation (15), Art. 397, so that we have to consider these three terms only. 422. By equation (3), Art. 377, we have *yxo tyyo cix 486 CALCULUS OF VARIATIONS. Moreover, by equation (4), Art. 378, we have +eAo / W'^^-^^^-^+eAo / y^^y^'^r)^^^^^' (4) PXi tyo px^ lyo -i. / ^^yVrSqdx-J^^ I y,{y,Vr),Szdx. (5) Now we must observe that every y^ refers to the contour only, and hence it varies with x, but is independent of the general values of jf. Hence {Vry.y = Vry,.^ F,% (F,j,X =yxVrr (6) Substituting these values and collecting results, we may write L L Vr^rdydx^J^^ l^ \Vry..+ 2Vr%+Vr,{y.r\Szdx +X L Vr{y.rSgdx-l^ J^^ Vr'S.dy+l^ l^ V^Spdy -L L ^'ry.^'^L Jy, Vr'Szdydx. (7) Again, by equation (3), Art. 377, we have rf^v/l?-dydx = n^x PVx /«i PVx /»^i fVx -L X V^'S'^dydx + l^ l^ VMdy-l^ l^ V.y.Sgdx.{Z) Kio) EXTENSION OF SARRUS' S METHOD. 487 By equation (5), Art. 379, we have X'X" ^-'^''y '^^ -£: ly^'^d'' ■' (9) and by equation (6), Art. 380, we have /'X>-.f*=-/T'"..''*+/7."".'- -/'•r'-.f*=/"x>.."*-/7:''.*- Hence, collecting results, we have Lastly, by equation (5), Art. 379, we have - Vt,-^-dydx^ / Vt.Szdydx- I Vt^^zdx. Whence 488 CALCULUS OF VARLATLONS. Adding these results, and also the second member of equation (15), Art. 397, we finally obtain + / / VDydx-\- / VDxdy ' 4-23. Now when ^is to be a maximum or minimum, we must, as before, have M — o irrespectively of the conditions which are to hold at the' limits. This equation, which must subsist for every point of the required surface, will be in gen- eral of the fourth order, and its solution, when any exists, will not contain more than four arbitrary functions. Next, if in the terms at the limits we regard the quantities Sz, Sp^ dq, Dy and Dx as independent at each limit, we shall evidently obtain the following system of equations : (14) Vry„ + Vr,{yif + (2 F/ - V^)y^ +V,-V/-Vt, = o, Vr(.yxY-Vsy:r+Vt=0, Vp-V/-Vs, = o, F, = o, (15) F.-F-^. = o. (16) V^o, J^^ Vdy=o: (17) EXTENSION OF SARRUS'S METHOD. 489 where (14) and the first of (17) hold along the upper and lower contour, (15) and the second of (17) along- the left and right portions when they exist, while (16) holds only for the four corners, or the junction of the different portions of the con- tour, the differentials jFa:, etc., at these points being taken with reference to that one of the two intersecting portions which we may happen to be considering. But under the present supposition the total number of equations at the limits would be (16), whereas we have at the most not more than four arbitrary functions with which to satisfy them ; so that, as before, we must impose some restric- tion upon the contour which will reduce the number of these equations. 424. If we suppose the form of the projected contour to be fixed, equations (17) will disappear, and we shall have but twelve equations at the limits ; and if, in addition, we suppose the left and right portions to be wanting, equations (15) and (16) will also cease to exist, and we shall have but four equa- tions at the limits. In this case, therefore, in which \}i\Q pro- jected contour consists merely of two curves which meet, we may reasonably suppose that a complete solution might be possible. We can render equations (14) somewhat more symmetrical. For differentiating the second, regarding 7 as a function of x, as indeed it is along the projected contour, we obtain + {Vt-Vsy.+ V/ = o. (18) Now from this equation we eliminate f^x by the first of equa- tions (14), obtaining an equation involving jx with its second and third powers. Then from this new equation eliminate successively (jxT and (fa^Y by means of the second of equations T4), the work presenting no difficulty whatever, except its 490 CALCULUS OF VARLATLONS. length. By these operations, and retaining the second of equations (14), we shall have -^\VtVr,^VlV^-V,-Vr')^Vr{V,'^ZV,-2V^)\dy^o, Vrdf- V,dydx-\- Vtdx'^o. (19) 4-25. It is easy to show that conditions (14), or rather (19), must hold also along the right and left portions of the con- tour when they exist. For since along these portions dx = o, the second of equations (19) will give Fi- == o ; so that F^^ = o, and then these three conditions will cause the first of equa- tions (19) to reduce to Fp-F/-F,,=:o. For the four corners of the required surface we merely join to equations (14) or (19) equation (16). We might in the same manner as before discuss the case in which the required surface is to be limited by any given surface or surfaces. But as this examination would not prove useful, because of the scarcity of actual problems, and as it is believed that the reader will now be able to investigate these cases for himself, we shall proceed no further in the discussion of this subject. 426, We have now seen that the method of M. Sarrus enables us to investigate in a systematic manner the condi- tions which must, under any supposition, 'hold at the limits in order that U may be a maximum or a minimum ; and so far as this method itself is concerned, it should be regarded as satisfactory and sufficient. But while it gives the conditions which must prevail at the limits, if there be any solution, it still remains for us to determine whether or not these condi- tions can be fulfilled, and we shall find at this point that the EXTENSION OF SARRUS' METHOD. . 49 1 theory is much less satisfactory than in the case of simple in- tegrals. For supposing V to contain differential coefficients of z to the order n inclusive, we know that the equation M^o will be in general a partial differential equation of the order 2n. Now we are rarely able to integrate an equation of this class, and are not certain that all such equations admit of any solution at all in finite terms ; and even if we suppose a solu- tion to exist, we cannot tell a priori how many arbitrary func- tions it must involve, all that we know being that the number of these functions will not exceed that which marks the order of the differential equation in question. Moreover, even if we knew the number of these functions, we could not say how many conditions ihey might be made to satisfy, since we would not know what should be the quantities under the functional sign. Also, when we have obtained an integral of one of these equations, we cannot be always certain that the solution is of the most general possible character. 427. From what has been said, it will appear that we can- not, as in the case of simple integrals, assert that because the equation iI/= o is of the order* 2??, the general solution can be subjected to 2n conditions at the limits ; although the ex- amination of particular cases, as well as the analogy of simple integrals, would lead us to infer such to be the case. If, for example, we require that the surface given by the equation M — o shall pass through 2n distinct curves, or shall have its edges upon 2n surfaces, we do not know that these conditions can be satisfied, but our inference that the}^ can is supported by the following additional considerations. In an equation of the form M = o we can assign arbi- trarily the values of z corresponding to x = o or to some function of x and y equals zero, and also those of the first 2n — I differential coefficients of z with respect to either vari- able, X for example. Now by assigning the values of z we compel the surface to pass through one given curve, which 492 CALCULUS OF VARIATIONS. would be all that we could do in the case of a partial differ- ential equation of the first order. When the equation is of the second order, we can, as before, make the surface first pass through some curve, and then, by suitably assigning the value of/, can fix the position of the tangent plane along this curv,e ; that is, can make the surface pass through two curves which are consecutive. In like manner, when the equation is of the order 2n, we can effect that the surface shall pass through 2n curves which are consecutive one to another ; and since this can be done so long as. the curves are indefinitely near one another, we may infer that it would also be possible if the curves were sepa- rated by finite spaces, although we must be careful not to speak with too much certainty upon this point. The last two articles are due chiefly to Moigno and Lin- delof. See their Calciil des Variations. 428. The equation M = o will not, however, always rise to the order 2n. If Fbe a function involving x, y, z,p and q only, thus naturally making M oi the second order, it is readily shown that M will not rise above the first order if V have the form V = fix.y, z) +flx,y, z)p +flx,y, z)q, (i) and in this case only. But if F contain x^y, z,p, q, r, s and /, giving usually M oi the fourth. order, it can be shown that to prevent M from rising above the third order, it is necessary and sufficient that, A, B, C, D and E being severally functions of x^y, z,p and ^, F shall be of the general form V=.A(rt-s')^Br-\-2Cs-^Dt-^E. (2) Moreover, it is shown that in both these cases the equation M —o cannot in reality rise above the order 2n — 2. See the work of Prof. Jellett, page 249. It will be remembered that the corresponding case for EXTENSION OF JACOB PS THEOREM. 493 simple integrals arises from the fact that the integral / ^Vdx is capable of some reduction by integration, and should be reduced before applying the calculus of variations. But we cannot extend the analogy. For in the present case no such reduction is, in general, possible. Section IV. EXTENSION OF JACOBPS THEOREM TO THE DISCRIMINATION OF MAXIMA AND MINIMA OF DOUBIE INTEGRALS. 429, We will now present a mere outline of the method of extending the theorem of Jacobi to double integrals, con- / sidering the case in which F is a function of x, y, z, p and q only, and supposing, as usual, that the limiting values of x, y and z are fixed. Now since the terms of the first order must vanish, if U is to become a maximum or a minimum, we shall have 6U=- r^ r^ \ V,M +2 V,p6z Sp + 2 F^3 Sz dq +2F^,(^/d^+Fpp^/+ V^^6q''\dydx,{i) Now we can change the form of (J C/" thus: 6U = \£'J^^\V,,Sz+V,,6p+V,^6q-{V,,Sz+V^p6p+V^,dgy - ( Vzq Sz + Fpg Sp + F^g Sq)^ ] Sz dy dx. (2) The truth of (2) can easily be verified by integrating once by parts each of the quantities within the accented pa- 494 CALCULUS OF VARIATIONS. rentheses, the first set with respect to x and the second with respect to y, remembering that the limiting values of z are fixed. Thus, for example, Proceeding thus with each term, we shall obtain the same form for (5' C/ as in (i). Now let Then (2) may be written - ( Vp.. /../.of q-py' ^^ V i^f^q' Sy I dx /^i f*y\ 1^0 c 4) , ) + / / / ) — - ^ 6z+Vi+p''+q''dx[dy, (5) 528 CALCULUS OF VARIATLONS. and it is evident that SA^, SB^ and SC^ will be expressed by precisely the same equations with every single suffix o changed to i. 458. But, as appears from (3), we must, to obtain dU, multiply the last three, or rather the last six, of equations (5) by — a, and add the result to Sv. If, then, in this equation we resolve all the signs of substitution, and then bring together the terms which contain like variations, and are affected by like substitutions (which M. Sarrus does), SU will consist of thirty distinct terms, six holding throughout the six limiting faces, and the remaining twenty-four referring to what might be termed the twenty-four edges of these six faces, each actual edge of the body being regarded as belonging to either of two faces ; and these thirty terms are independent. But following the device of Moigno and Lindelof, we may write SUin the following condensed manner: t/o-o t/2/0 Izo j \dx |/i ap + /+/ d dy ^-^j^p'^j^q )\ "'^ \\dsdydx -^T' T' T'dxdzdy SOLID OF MAXIMUM VOLUME, $2g +(/± VT+ynSx id^:^0, (6) where the signs of substitution denote throughout SW the same series of operations as before ; the sign ± in the first three terms denoting that at the first substitution the upper, and at the second the lower is to be taken ; the same sign in the last three terms signifying that the upper is to be taken when the quantities substituted have the same suffix, and the lower when they have not, while these results must still, as the sign — or + indicates, be multiplied by — i or + i> ac- cording as the quantity above the left-hand sign of substitution has the suffix i or o. But the reader who may prefer can easily write out the thirty terms from (5), and verify directly the last and the following assertions. 459. Now equating to zero the coefficient of d^ in the first term of (6), we have Or denoting by r and r' the principal radii of curvature, the last equation is equivalent to i + l=i and ! + -,= --; (7) r r a r r a 530 CALCULUS OF VARLATIONS. the first holding throughout the face C^, and the second throughout the face C^. Then the two equations (7) show that C^ and C^ have their mean curvatures constant, but turn their convexities in opposite directions. Equating to zero the coefficients of dz in the fourth term, we have '-^'' =Ti, (8) ni+/+/)(i+/o which involves the four equations relative to the intersection of the (7's and ^'s, the negative sign holding for the edges C^ B^ and C^ B^, and the positive sign for the edges C^ B^ and C^B^, But the first member of (8) equals the cosine of the angle made with each other by the two surfaces along their common intersection ; and since this cosine is unity, we infer that the ^'s and Cs are always tangent or accord along their common edges. We observe, also, that equation (8) will Cause the coeffi- cient of (^K in the same fourth term to vanish without giving any additional equations. For since the ^'s and ^^'s are tangent, / and q will have the same meaning in both along their intersection : thus ten equations have been considered. Equating to zero the coefficient of ^z in the fifth term of (6), we have, for the intersections of the ^'s and (7's, ^ =Ti; (9) 4/1+/+^' — I when the suffixes are alike, and + i when they are un- like. Equation (9) denotes that the faces C^ and C^ must, along their intersections with the planes A^ and A^^ be normal to the axis of X ; that is, they must be tangent to or accord with these planes. TJien we observe, as before, that (9) causes the coefficient of Sx in this same fifth term to vanish without giving rise to SOLID OF MAXIMUM VOLUME. 53 1 any additional equations. Thus, then, eight terms more have been caused to disappear. 460. Now having equated to zero the second term in (6), which is relative to the cylinders B^ and B^, and remembering that dy must remain constant along any particular generatrix, but is independent for each, we shall obtain i ^^1 V n ay ^ dz = o, (loj dx i/i-\-y which, with the positive sign, holds for any generatrix of B^, and with the negative sign for the corresponding generatrix of B^. But as the integration in (lo) is to be effected regard- ing X, y and y' as constant, we have ^•^ -..-..) = 0. (II) dx |/i -|-y Equating the first factor to zero, we would obtain a cylinder of radius a, the limits of z being wholly undetermmed. But neglecting for the present this supposition, we must have z^ = Zj] that is, B^ and B^ vanish or reduce to mere edges. The condition z„ = z^ will cause also the first member in the last term of (6) to vanish without giving any new equa- tions ; so that thus six more terms in all disappear. Equating to zero the third term in (6), and remembering that Sx^ and dx^ are 'two independent constants, we have r^ r'dzdy = o, (i2) an equation which involves two, as it holds for either of the faces A^ or A^, and shows at once that these two plane faces must also disappear. Then (12) will cause the last member in the last term of (6), which is relative to the four intersections of them's and 532 CALCULUS OF VARIATLONS. B'% to vanish without giving any new equations. Thus all the terms in (6) have been caused to vanish severally. 461. If we admit into the solution the cylinder with ra- dius a^ or for A^ and A^ any edge perpendicular to the axis of X (which is probably admissible), we cannot say that all the conditions of the question could be satisfied. But if we as- sume B^ and B^ to become mere edges, and A^ and A^ to become points only, the volume in question must be entirely enclosed by the curved faces C^ and C^, Moreover, from what has been shown it will appear that these two faces must be respectively perpendicular to the plane of xy along their common intersection, and they must there- fore meet in such a manner as to coalesce and to form one and the same surface, which will be given by the equation derived from (7), l-+a=y (■3) , Now the sphere of radius 2a will evidently satisfy all these conditions. But in order to exclude all other hypotheses, it would still be necessary to show that the sphere is the only admissible solution obtainable by equating to zero the terms of the first order in SU. But the proof of this fact has never yet been obtained by analyses ; and even if it could be, it would still be necessary to show that the sphere would cause the terms of the second order to become always positive, or else those of some other even order to become so, the preceding having reduced to zero ; and this would present a new and probably an insurmountable difficulty. Moreover, as we take the entire sphere, we shall be obhged to deal with some quan- tities which will become infinite ; which fact might of itself throw some doubt upon our investigations. But although the complete discussion of this problem appears to be beyond the power of the present methods of analysis, we are assured from other considerations that the sphere is its true and only solution. CHAPTER IV. APPLICATION OF THE CALCULUS OF VARIATIONS TO DETER- MINING THE CONDITIONS WHICH WILL RENDER A FUNCTION' INTEGRABLE ONE OR MORE TIMES. Section I. CASE IN WHICH THERE IS BUT ONE INDEPENDENT VARIABLE. Problem LXVIII. i 462. Suppose we seek by the calculus of variations to maxi- mize or minimize the expression Then returning to our former notation, we shall have dx y / y d'Q-n"- _^ ^y" I ^^y\ 534 CALCULUS OF VARLATIONS. SO that the equation M=N-P'-\-Q"z=:0 will reduce to o = o ; that is, M will vanish of itself, or identi- cally ; so that we obtain no equation from which we can derive a general solution, and have left merely the terms at the limits, which may be written 4-63. Now in seeking to explain this anomaly, we observe that Fmay be written y{xy" -\ry')-xyy' f Whence we see that fvd. = ^-l+c, and U = iy-l. (3) Thus it appears that U can in this case be freed from the sign of integration, and that the discussion of the conditions which will render it a maximum or a minimum does not, strictly speaking, belong to the calculus of variations ; and we can readily show that whenever U is integrable, M must vanish identically. For assume the equation U—J^ Vdx, where V is any function of x, y, y\ ^ -^ . . y^'^\ but is of such a form that Vdx shall be immediately integrable ; that is, in- tegrable independently of any relations which may hold be- tween X and y. Then we know that by definite integration U may be written ^-l> _ x,y,y,.,..y(^-'^)), (4) CONDITIONS OF INTEGRABILITY. 535 which shows that U depends solely upon the limiting values of X, y, y, etc., the relations between x and y being altogether in our power. Now if in U^ before integration, we vary j^,j/', etc., but suppose these quantities at the limits to remain fixed, Z7will undergo no change; that is, SU will vanish; and be- cause ^y, dy\ etc., are zero at the limits, ^^can, as usual, be reduced to the form SU=£-MSydx^o, (5) to satisfy which, since Sy is in our power, M must vanish. But unless M vanish identically, we shall, by equating it to zero, have an equation which, if it be integrable, will deter- mine J/ as a function of x, or, if not integrable, will establish an implicit or differential relation between them, both of which are contrary to the conditions of the question. If, therefore, Vdx be integrable immediately — that is, without assuming any particular relation, either explicit or implicit, between x and y—M must vanish identically. 464. Conversely, if, Z7and F having the same meaning as before, we fin'd M to vanish identically, we may conclude that Vdx is immediately integrable. For we see that SU will in this case consist of the terms at the Hmits only, as in equation (2), so that we infer that C/must depend solely upon the values which X, y^ y' , etc., may have at the limits ; and hence that U must in reaUty be a function of these quantities only, which, so long as y is wholly in our power, could not be true unless Vdx were immediately integrable. This mode of reasoning would seem to be sufficiently con- clusive ; nevertheless it is not so regarded by Prof. Tod- hunter, and the reader will find in his Integral Calculus, Art. 382, an attempt at a more rigorous demonstration. 53^ CALCULUS OF VARIATIONS. Problem LXIX. 465. V having the same form as before^ it is required to de- termine the conditiofis which will render V immediately integrable any num,ber of times, m. First assume vt to be 2, and we have f I fvdx \ dx = xjvdx -Jx Vdx ; (i) and hence, to insure that F shall be twice immediately inte- grable, we must have both F'and Vx immediately integrable; and conversely, if these quantities be immediately integrable once, Fwill be immediately twice. Now the first condition will give N- P'-\- Q" - etc. = J/= o, (2) which must be true identically ; while putting v for Vx, the second condition will give, in like manner, Vy — Vy/-\- Vyn "— CtC. == O, ^ (3) which must also be true identically. But (3) may be replaced by another equation, thus : Vy—X Vy, VyI = X Vyf , Vyt, = X Vy>f , CtC, Vy/= X Vy/+ Vyr, Vy.' ^ = X Vy.' ' + 2 Vy.\ Vy./" = xVy,.'"+2>Vy./', etc. Substituting these values in (3), and omitting those terms which are known to be zero by (2), we shall obtain P-2Q-\-zR"-Qic. = o, (4) which must be true identically. CONDITIONS OF INTEGRABILITY. 53/ Hence that Fmay be immediately integrable twice, equa- tions (2) and (4) must be identically true. Now, in the more general case in which fn is any number less than n^ it is generally shown in works on the integral calculus that, if we denote by U the result of the integration oi V m times, we may exhibit (7 thus : U= — ^ — \ ^"^-1 fvdx — (m— i)x-^-^ fxVdx 1 {in — i){m — 2) 1-2 x^-^fx''Vdx-ttc.±fx^-^Vdx\. (5) Whence it appears that to render F integrable //^ times it is necessary and sufficient that the quantities V, Vx, Vx"", .... y^m-i shall be severally integrable ; and the equations arising from these conditions can be determined precisely as before. Thus if m be 3, we shall find, in addition to equations (2) and (4), the identical equation e-^i?' + i^5''-etc. = a (6) 1-2 1-2 ^ ^ Problem LXX. 466. // is required to determine the conditions which will render Vdx immediately integrable, V being any function of x and the dependent variables y and z, together with their differen- tial coefficients with respect to x ; that is, y\ y" , z' , z" , etc. Putting, as before, U for the integral, and transforming SU, we shall obtain, as in Art. 303, a result which may be written SU^L,-L,^ r^M dydx + r^YSz dx, (i) where M= Vy- Vy/+ Vy>/'- ctc, N = V, - F/+ F,//'- etc. (2) 538 CALCULUS OF VARLATIONS. Now, as before, we may suppose the limiting values of j/, y, z, z', etc., to be fixed, so that L^ and L^ will vanish. Moreover, Sy and Sz are entirely independent, so that M and TV must severally vanish if C/is to depend solely upon the limiting values of x, y, y', z, z', etc. But either or both the equations M —o and N — o, unless they be identically true, would enable us to establish some explicit or implicit rela- tion between x, y and z, whereas we require that Vdx shall be integrable irrespectively of any such relation, other than that jj/ and z are to be regarded as functions oi y and x. If V were integrable m times, it is easy to see that, as in Prob. LXIX., we must have F, Vx, Vx\ etc., immediately integrable, since equation (5) of that problem requires merely that V shall be a function of x, and it might, therefore, con- tain any number of dependent variables, y, z, ti, and their dif- ferential coefficients with respect to x. Hence we should evidently obtain with such equations as (2), (4) and (6) similar equations in z. Moreover, it will appear that for any other dependent variable ti which V may contain, we shall require in addition a similar set of equations in u. Section II. CASE IN WHICH THERE ARE TWO INDEPENDENT VARIABLES. Problem LXXI. 467. It is required to determine the conditions ivJiich will render I I Vdy dx reducible to a single integral, ivhere V is any function of x, y, z, p and q, x and y being tzvo independent vari- ables, and p and q partial differentials of z with respect to these variables. Denoting the definite integral by U, we know that after transformation (JC/may be written CONDITIONS OF INTEGRABILITY. 539 dU=L +fJ'fJ'M Ss dydx, (i) where Z, although consisting of simple integrals, involves only terms which relate to the limits of the integration ; and by supposing z to be unvaried along the lines x — x^ and x = x^, we can make L consist only of quantities which are functions of X, and the variations of such quantities. Now we know that if we regard all the quantities at the limits as fixed, L will vanish, so that if U can be reduced to a single integral depending upon these quantities only, 31 must vanish ; and if this reduction is to be possible without deter- mining z as some function, explicit or imphcit, of x and y, M must vanish identically, otherwise the equation M — o will establish some such relation. On the other hand, if J/ vanish identically, (^^ will reduce to Z, and we infer that, as it depends solely upon quantities at the limits, U is immediately reducible to a single integral. 468. Now we can determine what form Fmust have to render this reduction possible. For M=^V,- F/- F,,; (2) and if M is to vanish identically, it is evident, in the first place, • that all terms containing/, q, r, j and / must vanish. But we have seen (Art. 428) that when, and only when, Fis of the form/j +/2/ 4-/3^, the equation M —o will fail to rise above an order n — 2\ that is, 2 — 2. Such, then, must be the form of F; but that J/ may en- tirely vanish it will be necessary, in addition to this, that/j, f^ and /g, which are all functions of x, y and z only, shall be subject to a certain additional relation. For we have V^=j-^' ^P^f^' v^=U 540 CALCULUS OF VARIATIONS. SO that the equation M = o, after rejecting terms containing differential coefficients of z, gives dz dx dy ^ and the /'s must be so related as to render this equation also identically true. 4-69. Although not very rigorously demonstrated, the foregoing are all the leading theorems relative to this subject, and it would be unprofitable to pursue it further. For while the calculus of variations gives us the means of determining whether or not V is immediately integrable, it does not of itself indicate the method of effecting the integration; and this method is what we wish chiefly to know. The theorems given in the preceding problems relative to this subject, which is often called the theory of integr ability, or the conditions of integr ability, can be established without the aid of the calculus of variations, but less easily. The reader who may wish to pursue this subject further is referred to the treatise on the calculus of variations by Prof. Jellett, Chap. X., and also to Todhunter's History of the Calc. of Van, Chap. XVII. CHAPTER V. HISTORICAL SKETCH OF THE RISE AND PROGRESS OF THE CALCULUS OF VARIATIONS. 470. Questions of maxima and minima were among the first to occupy the attention of mathematicians after the in- vention of the differential or fluctionary calculus, which, according to Woodhouse, occurred about the year 1684, or three years prior to the publication of the Principia. The ordinary calculus was not, however, given to the world at once in a single treatise, but was developed gradually in essays, in communications to learned societies and journals, and in letters between men of science. The first question considered, of that particular species of maxima and minima which forms the chief subject of the cal- culus of variations, appears to have been that of the solid of minimum resistance ; and this was first proposed by Newton in the Principia. But although Newton was the first to con- sider a question belonging to the calculus of variations, no importance seems to have been attached to this problem either by himself or his contemporaries, and it did not become at that time the subject of discussion. The true beginning of our science dates from the month of June, 1696, when John Bernoulli, Professor of Mathema- tics at Groningen, proposed in the Acta Eruditorum (or Doings of the Learned), a work then published at Leipsic, and at that time the chief medium of communication between men of science and letters, the following problem : 542 calculus of variations. "Problema Novum* "Ad cujus solutionem mathematici invitantur. " Datis in piano verticali duobus punctis A et B, assignare mobili J/viam AMB, per quam gravitate sua descendens, et moveri incipiens a puncto A, brevissimo tempore perveniat ad alterum punctum B.'' This problem engaged at once the attention of Leibnitz, James Bernoulli, brother of John, and Professor of Mathe- matics at Basle, and the Marquis de I'Hopital, the first two of whom appear to have solved the problem within the allot- ted time, which was six months. Leibnitz at once forwarded his solution to the proposer, asking that it might not be imme- diately published, in order that other mathematicians might be encouraged to attempt the problem ; and he subsequently, as no solution appeared, requested that the period might be extended, a request with which John Bernoulli compKed, and accordingly reannounced the problem in a programme dated at Groningen, Janua;-y, 1697. Upon learning of this exten- sion, James Bernoulli retained his solution, being desirous, as he stated, of investigating and adding to the problem certain others of a similar character ; which he did, as we shall subse- quently see. In the Acta for the following May were published the solu- tions of the two Bernoullis, together with one by De I'Hopi- tal, the last being without demonstration. James is in advance of his brother ; but as his solution is given by Woodhouse, it will here suffice to say that both brothers assume the prin- ciple that whatever maximum or minimum property is pos- sessed by the whole of any required curve must be possessed also by every portion of the curve ; and that therefore, if * " A New Problem, to the solution of which mathematicians are invited. " Given two points A and ^ in a vertical plane, to find for the movable (par- ticle) M, the path A MB, descending along which by its own gravity, and begin- ning to be urged from the point A, it may in the shortest time reach the other point B.'' HISTORICAL SKETCH. 543 we consider the required curve as a polygon of an infinite number of sides, it will be sufficient to consider two consecu- tive sides or elements. But this principle, while it enabled them to obtain in this case a correct result, is not universally true. 47 (. At the close of his paper James Bernoulli proposed two additional problems : first, to determine the curve of quickest descent from a given fixed point to a given vertical line ; and second, among all curves having a given length and a given base, to find the curve such that a second curve, each of whose ordinates is some function of the corresponding or- dinate or arc of the first, may contain a maximum or minimum area. But although in the first of these problems we have a particular case of the question subsequently considered by Lagrange as to what conditions must hold at the limits in maximizing or minimizing a definite integral, little appears to have been effected in this direction prior to the re- searches of that mathematician ; so that we shall follow the second problem only. The second case of this problem led to an acrimonious discussion between the Bernoullis which was little creditable to John. For still adhering to the principle mentioned at the close of the last article, which in this case fails, owing to the isoperi metrical property that the required curve must have a given length, he continually obtained erroneous re- sults ; nor would he frankly acknowledge his error after it had been indicated by his brother. No solution of this prob- lem, however, appeared until James BernouUi, in May, 1701, published his in the Acta; although a solution without demon- stration had appeared in the Acta for the preceding June. In this demonstration three consecutive elements of the required curve are taken instead of two, and a rude mode of imposing the isoperimetrical condition is shown. A solution much more finished, but evidently borrowed from that of his 544 CALCULUS OF VARLATIONS. brother, was subsequently published by John Bernoulli in the Memoirs of the Academy of Science for 1718 ; and this solution may be found in Woodhouse's Isoperimetrical Problems. But no further advance, worthy of notice in a sketch like the present, appears to have been made until the advent of Euler. 472. The first contribution of this mathematician to our science was a memoir published in the sixth volume of the Ancient Commentaries of Petersburg, 1733. In this memoir Euler, taking up the subject where it had been left by the Bernoullis, divided his problems into classes : the first including those of absolute maxima and minima, the second those prob- lems of relative maxima and minima in which but one restriction is imposed upon the variations, as in the problem of the bra- chistochrone when the path is to have a given length ; while the third included those relative problems in which two re- strictions are imposed, as when the brachistochrone path is to have a given length and also to enclose, with the aid of its extreme ordinates and the axis of x, a given area. The erro- neous principle that the maximum or minimum property of the whole curve belongs to each portion also was virtually adopted, two consecutive elements only being considered in the problems of the first class, three in those of the second, and four in those of the third. Nevertheless, as he proceeded, he established and tabulated formulse — twenty -four in all — for the various cases which might arise ; and by this means was, at the close of this memoir, much in advance of the Bernoullis. About the year 1740 or 1741 Euler summed up his re- searches in a second memoir published in the eighth volume of the Commentaries of Petersburg, the date of the volume being 1736. But this date proves nothing, as the same volume contains observations made in 1740. Euler had now discov- ered that when F is a function of x, y, y,. . . . y^^\ his previous formulae might be expressed by one more general formula. HISTORICAL SKETCH. 545 which is still in use, and which we have denoted by the equa- tion M = o. Also, the principle that the maximum or minimum property of the whole curve will belong equally to every portion was examined, and shown, in some cases at least, to be untrue ; and lastly, some advance seems to have been made in the treatment of problems of relative maxima and minima. By this memoir the calculus of variation was greatly im- proved, and it contained, in fact, nearly all that its author ever discovered relative to this subject, although in a very confused and ill-arranged form. In 1744 Euler published a tract entitled Methodiis inveni- endi Lineas curvas Proprietate maxwii mininiive gauderitesJ^ This work, which was the most famous of its author re- lative to this subject, displa3^s an amount of mathematical skill almost unrivalled. Problems were here, as at present, divided into two great classes, absolute and relative, and the treatment of the second was for the first time reduced to a perfect science by the discovery of the artifice still employed, and termed the Method of Euler. This work is also general- ly clear and systematic, containing an abundance of examples, including, with many others, most of those given by us m our first chapter ; and at its close Euler had carried our science so far beyond the point which had been reached by the Bernoullis that he may, almost equally with Lagrange, be regarded as the author of the calculus of variations.f Euler subsequently published in the tenth volume of the New Commentaries of Petersburg, i 'j66, two memoirs ; in the first of which, entitled Elejiienta Calculi Variatiouum (or * " Method of finding curved lines enjoying the property of maximum or minimum." f The preceding account has been taken almost entirely from Woodhouse's Isoperimetrical Problems; but for what follows we are indebted chiefly to Todhunter's History of the Progress of the Calculus of Variations during the Nineteenth Century. 54^ CALCULUS OF VARLATLONS. Elements of the Calculus of Variations), he first gives our science its present name ; while in the second he enunciates the theorem of Prob. LXVIII. : this being apparently the first investigation ever made relative to the conditions of in- tegrability. Subsequently in his Integral Calculus, 1770, he extended the theorem to two dependent variables, as in Prob. LXX.; while Lexel, in 1771, established the principle of Prob. LXIX. 473. Prior to this period Lagrange, w^ho is commonly re- garded as the author of the calculus of variations, had com- menced his labors. But as we have not space to consider his writings in detail, we shall merely indicate the particulars in which he improved our science. First. Much ambiguity and awkwardness had previously arisen from the want of a good method of distinguishing be- tween ordinary differentials and those differentials or incre- ments which we now call variations. This difficulty Lagrange overcame by the invention of the symbol d, which, like dy could denote either an increment or an operation, and proved of the highest importance. Secondly. The formula M =z o, and others, had been de- rived by Euler from geometrical conceptions by breaking the integral, or the required curve, into parts, and operating labo- riously upon two, three, or four consecutive elements. But Lagrange, by deriving these formulae by the methods now in use, shortened the processes of obtaining them, and placed our science upon its true analytical basis. Thirdly. The formulae of Euler determined merely the nature of the required curve, its extremities being supposed to be fixed. But Lagrange, in what he termed the definite equations, first gave the form and the interpretation of all those formulae which are still employed when the extremities of the required curve are not fixed, and which we have called the equations or terms at the limits. HISTORICAL SKETCH. S47 Fourthly. Lagrange invented that general method which is still employed, and known as the Method of Lagrange, and which enables us by the use of one or more indeterminate multipliers to discuss those cases in which the variables are connected by an implicit relation merely ; that is, by a differ- ential equation which is not integrable. Lastly. Lagrange first attempted to extend the calculus of variations to the case of double integrals. This he did by discussing Prob. LVIL, obtaining our equation (lo), Art. 366, without considering the terms at the limits. 4-74. In the year 18 10 appeared an English work, entitled " A Treatise on Isoperimetrical Problems and the Calculus of Variations," by Robert Woodhouse, A.M., F.R.S., Fellow of Caius College, Cambridge. The first five chapters of this work, which is a small octavo of 154 pages, with 9 pages of preface, are devoted to a careful history of the subject to the time of Lagrange, and are all that are now of any in- terest, the remaining three containing little history of im- portance. The subject having been next, but not very successfully, treated by Lacroix in his Traits du Calad Diffei-enticl et du Cal- cul Integral, second volume, second edition, 18 14, there ap- peared two German works. The former, by E. H. Dirksen, which is a small quarto of 243 pages, with 8 pages of preface, was published at Berlin in 1823, and is entitled "Analytical Exhibition of the Calculus of Variations, with the Applica- tion of it to the Determination of Maxima and Minima." The latter is entitled " The Theory of Maxima and Minima," by Dr. Martin Ohm, Berlin, 1825, and is an octavo of 330, with a preface of 18 pages. None of these works, however, extended the calculus of variations, and we now resume the history of its progress. 475. The first discussion of the discrimination of maxima and minima appears to have been undertaken by Legend re, 548 CALCULUS OF VARIATIONS. who, about the year 1787, elaborated the method already men- tioned in Art. 187, and which was published the following year in the History of the Royal Academy of Science. This method was subsequently adopted by Lagrange, although he indicated the defect noticed in the above article. This method is explained in Todhunter's History of the Calculus of Varia- tions. Legendre seems also at the same time to have given the first instance of a discontinuous solution by showing in the discussion of Prob. XV. that it might be necessary for the re- quired curve to be in part rectilinear. In the Memorie delV Istituto Nazionale Italiano^ Vol. II. Part II., Bologna, 18 10, Brunacci extended the method of discrimination to the case of a double integral ; and although his method is open to the same objection as that of Legendre for single integrals, he succeeded in establishing all the con- ditions relative to Fpp, f^g and F^g mentioned in Art. 431; and their discovery appears to be due to him. 476. The variation of a double integral when the limits are also variable, the exhibition of the terms at the limits so as to determine the conditions which must there hold, and the vari- ation of a multiple integral in general, were subjects which had not yet been investigated, and they next engaged the at- tention of mathematicians. Three memoirs were pubhshed bearing more or less directly upon these subjects : the first by C. F.Gauss in 1829, the second by Poisson in 1 831, and the third by Ostrogradsky in 1834. But while these writers effected much, they did not succeed in determining in a general manner the number and form of the equations which must subsist at the limits in the case of a double or triple integral. 477. In the seventeenth volume of Crelle's Mathematical Journal, 1837, appeared a memoir, entitled "On the Theory of the Calculus of Variations and of Differential Equations," by HISTORICAL SKETCH, 549 C. G. Jacobi. This memoir, which purports to be an extract from a letter to Professor Enke, is devoted partly to the cal- culus of variations and partly to dynamics. In the first part Jacobi elaborated, but without demonstration, the theorem which bears his name ; that is, he assumed the truth of our lemmas, not even giving the forms of the functions A, A^, A^, B^, etc., although he determined the form of u, and merely touched upon the connection between u and v. See Art. 174. This brevity rendered the theorem the subject of numerous commentaries, as we shall presently see. Jacobi also touched upon the mode of transforming the terms of the first order in the variation of a double integral, but effected nothing of importance. 4-78, In 1 841 were published three memoirs relative to Jacobi's theorem : the first two by V. A. Lebesgue and C. Delaunay, in the sixth volume of LiouwiWe's /oicrna/ 0/ Mat ke- mattes, and the last by Bertrand in Xh^ Journal de V Ecole Poly- techniqite. The proof given by Delaunay is that which we have followed in our notes to Lemmas I. and II., and he has been generally followed by subsequent writers. 479. As, notwithstanding the labors of Gauss, Poisson, Ostrogradsky and Jacobi, no general method of treating the terms at the limits in the case of multiple integrals had yet been discovered, the Academy of Science, Paris, 1842, pro- posed for its mathematical prize the following subject: To find the limiting equations which must be combined with the indefinite equations in order to determine completely the maxima and minima of multiple integrals ; the formulas to be applied to triple integrals. Of the four memoirs presented, that by Sarrus was ad- judged worthy of the prize, while that by Delaunay received honorable mention ; the examiners being Liouville, Sturm, Poinsot, Duhamel and Cauchy. The memoir of Sarrus is entitled Recherches sur le Calcul 550 CALCULUS OF VARLATLONS. des Variations^ and may be found in the tenth volume of the Savants Etr angers, 1846, and occupies 127 quarto pages. By means of his new symbol of substitution, Sarrus may be said to have solved the problem proposed by the Academy, and his memoir is one of the most important contributions of the century. But this sign of substitution as invented by Sarrus, besides having an inconvenient form, signified merely the sub- stitution of a particular value of a variable for its general value, and his method therefore lacked brevity. The treatment by Delaunay is much less general, assuming that in the case of double integrals the limiting cylinder or surface is to be continuous and closed. His memoir was pub- lished in the 29th cahier of the Journal de V Ecole Polytech- niqiie, dated 1843, ^^^d seems to have been followed by all the writers on the calculus of variations subsequent to Moigno and Lindelof. 4-80. The next advance was made by Cauchy in a memoir on the calculus of variations published in the third volume of his Excrcices d' analyse et de Physique Mathhnatiqiie, 1844, extending^ from page 50 to page 130. This memoir is little else than a reproduction of the investigations of Sarrus, but in it Cauchy effected much of the needed condensation by giving to the sign of substitution, like that of integration in a definite integral, the power of denoting substraction also, while its form was changed to that which we have adopted. For further particulars regarding this part of our subject the reader who does not wish to examine the original memoirs may consult the chapters on Sarrus and Cauchy in Todhun- ter's History of the Calculus of Variations. 48(, We must next notice some systematic treatises which now appeared. As a successor of Woodhouse there appeared " A Treatise on the Calculus of Variations," by Richard Abbatt, London, 1837. This is an octavo of 207 pages, with 11 pages of pre- HISTORICAL SKETCH. 551 face, but is of no great importance at the present day, and could hardly be regarded as a complete treatise. In the year 1850 appeared a work entitled '' An Elementary Treatise ori the Calculus of Variations," by the Rev. John , Hewitt Jellett, A.M., Fellow of Trinity College and Profes- sor of Natural Philosophy in the University of Dublm. This work, which is an octavo of 377 pages, with an introduction and preface of 20 pages, is one of the most important which have appeared in any language, and is not elementary as its title would imply. But Prof. Jellett had not, as he himself tells us, been able to peruse the memoir of M. Sarrus, while that of Cauchy is not mentioned by him at all. Hence his dis- cussion of multiple integrals, in which he follows that memoir of Delaunay which received honorable mention by the French Academy, is defective, and cannot be recommended to the student. Ohm's treatise was succeeded by a voluminous work by Dr. G. W. Strauch, entitled Theorie und Anwendung des soge- nannten VariationscalcuV s^ Zurich, 1849. This treatise consists of two closel}^ printed large octavo volumes, the first contain- ing 499 pages, with 32 pages of preface, and the second 788 pages ; and is chiefly valuable for its great number of carefully solved examples, and historical notes, although, as might be expected, much of the matter has little or no connection with the calculus of variations. Strauch does not exhibit the theorem of Jacobi, although he generally examines the terms of the second order, employing the method of Legendre and Lagrange without even noticing its defect. He is also like Jellett deficient in the treatment of multiple integrals, not fol- lowing the method of Sarrus and Cauchy. Strauch sub- sequently, in 1856, presented to the Academy of Sciences in Vienna a memoir entitled Anwendujig des sogenannten Varia- tionscalcuV s auf zzvcifache und dreifache Integrale, and pub- lished in the i6th volume of the Denkschriftc7i of the Acad- emy, 1859, where it occupies 156 large quarto pages; and in 552 - CALCULUS OF VARLATLONS. this memoir he even declares that Sarrus and Cauchy did not solve the problem, proposed by the French Academy. His own memoir is, however, of no importance. In a few years appeared another German w^rk by Dr. Stegmann, entitled Lehrbuch der Variationsrechnung und ihrer . Anwendimg bei Untersuchungen fiber das Maxiinum und Mini- mum^ Kassel, 1854. This is an octavo of 417 pages, with 16 pages of preface, but is not so rich in examples as is the treatise by Strauch, while it possesses the two defects men- tioned in connection with that treatise. Prof. Bruun published in the Russian language " A Manual of the Calculus of Variations," Odessa, 1848, which is, accord- ing to Prof. Todhunter, an octavo of 195 pages. We may mention, finally, that Prof. Price in the second volume of his Treatise on Infinitesimal Calculus, Oxford, 1854, devoted more than 100 pages to our science, explaining the theorem of Jacobi, and touching upon the subject of double integrals. 482. After the publication of the three memoirs mentioned in Art. 478, the subject of the discrimination of maxima and minima was not considered for about ten years, after which it was resumed earnestly by mathematicians in papers, some of which we will next mention. In the third volume of Tortolini's Annali di Scienze Mathe- matiche e Fisiche, 1852, appeared an article of more than 40 pages by Prof. G. Mainardi, claiming, but without good rea- son, to exhibit a new method of discriminating maxima and minima. But he also extended Jacobi's theorem to double integrals, and his method has been followed by us in treating this subject. In the same volume appeared a short article on the same subject by Prof. F. Brioschi. Mainardi had indicated the value of the theory of determinants in connection with the exhibition of the terms of the second order, and Brioschi HISTORICAL SKETCH. 553 employed it freely, this being apparently the first attempt to apply determinants to this subject. There next appeared a quarto pamphlet of 20 pages regarding Jacobi's theorem, entitled Untersiichungen ilber Varia- tions-re chnung. Inaugural-Dissertation von Dr. Friedrich Eisen- lohr, Manheim, 1853. The subject was next considered in a work entitled " On the Criteria for Maxima and Minima in Problems of the Cal- culus of Variations," which was presented by Spitzer to the Academy of Sciences at Vienna in 1854. This work consists of two memoirs occupying together more than 135 pages, the first being published in the 12th and the second in the 14th volume of the Sitziingsberichte of the Academy, and to these memoirs we are indebted for the exceptions which we have noticed in connection with Jacobi's theorem. But Mainardi and Spitzer did not confine themselves to the development of Jacobi's theorem, but sought rather to establish new methods of their own, both of which are, according to Prof. Todhunter, " Legendre's method improved by additions borrowed from Jacobi." In the 54th volume of Crelle's Mathematical Journal^ 1857, appeared a memoir by Otto Hesse, entitled " On the Criteria for the Maxima and Minima of Single Integrals," extending over pages 227-273. Hesse confines himself exclusively to the application of Jacobi's theorem to smgle integrals involv- ing only one dependent vajriable, and his memoir is the most elaborate which has yet appeared regarding this subject. See Arts. 184, 186. In the 55th volume of Crelle's Mathematical Journal, 1858, appeared a memoir by A. Clebsch, entitled '' On the Reduction of the Second Variation to its Simplest Form," and extending over pages 254-273. The object of Clebsch was to general- ize the theorem of Jacobi, and to supply investigations like those of Hesse for the case in which the single integral con- tains several dependent variables with or without connecting 554 CALCULUS OF VARLATLONS. equations, and also for multiple integrals. The former point had not, so far as the author knows, been hitherto discussed, but the latter had been considered by Mainardi. The subject of multiple integrals is resumed by him in a third memoir, entitled '* On the Second Variation of Multiple Integrals," and published in the 56th volume of Crelle's Mathematical Jojir- naly 1859, where it extends over pages 122-148. His second memoir is " On those Problems in the Calculus of Variations which involve only one Independent Variable," and is in the same volume which contains his first memoir. 483. We now come to a most valuable work, enti- tled ''A History of the Progress of the Calculus of Varia- tions during the Nineteenth Century," by I. Todhunter, M.A., Fellow and Principal Mathematical Lecturer of St. John's College, Cambridge. Macmillan & Co., London, 1861. This volume is a large octavo of 530 pages, with 10 pages of preface, and, taken together with the first five chapters of Woodhouse, furnishes a complete history of our subject. But in addition to the mathematics necessary to the histori- cal sketches, much of which has been superseded by bet- ter methods, Prof. Todhunter has frequently introduced these better methods, and has given such other investigations of his own that his work contains nearly all the matter necessary to form a modern treatise, although, from the nature of the case, it is so arranged as to be of little service to the reader who is not already tolerably familiar with the calculus of variations. We append the subjects of the seventeen chapters : Chap. I., Lagrange, Lacroix ; II., Dirksen, Ohm ; III., Gauss ; IV., Pois- son ; v., Ostrogradsky ; VI., Delaunay ; VII., Sarrus ; VIII., Cauchy ; IX., Legendre, Brunacci, Jacobi ; X., Commentators on Jacobi ; XL, On Jacobi's Memoir ; XIL, Miscellaneous Memoirs ; XIIL, Systematic Treatises ; XIV., Minor Treatises ; XV.', XVI., Miscellaneous Articles; XVII., Conditions of Integrability. The last chapter is a complete history of the HISTORICAL SKETCH, 555 subject from the earliest times as it had not been mentioned by Woodhouse, nor had its history been given by any pre- vious writer. 4-84. A few months subsequently, but during the same year, 1861, appeared the last systematic treatise, the Calcul des Variations, by Moigno and Lindelof. But the title-page of this work, which is a small octavo of 352 pages, with 20 addi- tional pages of preface, introduction, etc., presents it in the beginning as merely the fourth volume of the Lemons de Calcid Differentiel et de Calcul Integral, by M. I'Abbe Moigno, the distinctive title following subsequently. According to Moigno, the chief credit of this work, which is the only complete treatise in the French language, belongs to his colleague, M. Lindelof, then a. young professor from the university of Helsingfors in Finland, who had made the calculus of varia- tions a specialty, and who gave Moigno freely the benefit of his knowledge. This treatise was the first to present a satisfactory account of the conditions which must hold at the limits when we wish to maximize or minimize the double or triple integral. But although the methods followed are substantially those of Sarrus and Cauchy, the authors have, in many cases, greatly simplified the formulae of their masters ; and to this portion of the Calcid des Variations the present author is almost entirely indebted for the discussions which have been presented in Chapter III. ; although the view of variations adopted by Moigno and Lindelof is that followed by Sarrus, and which has been explained in Section VL Chap. IIL 485. It had long been known that a discontinuous solu- tion might become necessary in certain problems. But although particular cases had been discussed by Legendre and others, nothing resembling a general theory of such solutions had yet been propounded. In the Philosophical Magazine for June, 1866, Prof. I. Tod- 55^ CALCULUS OF VARLATIONS. hunter first announced the principle that variations might be of restricted sign, thus rendering it unnecessary for the equation M — o \.o hold throughout C/; and this may be regarded as the fundamental principle of the theory in ques- tion. This discovery appears to have been due mainly to the difficulties presented by the consideration of Prob. XVI. In 1869 this subject was proposed at Cambridge for the Adams Essay, and elicited from Prof. Todhunter in 1871 the prize essay, which, with slight alteratfon, was published in the same year by Macmillan & Co., under the title " Researches in the Calculus of Variations." This work, which is an octavo of 278 pages, and 8 pages of introduction, is certainly the most important original con- tribution which our science has received since the appearance of the essay of Sarrus, inasmuch as, -in it, the author, while discussing incidentally many other points of interest, did for the theory of discontinuous solutions, what Sarrus did for that of multiple integrals. The case of single integrals only is dis- cussed, and these are, with a few exceptions, supposed to involve but one dependent variable. The theory is, how- ever, abundantly illustrated by examples ; and we cannot too strongly recommend the work to our readers, since, from it, we have derived most of what we have presented in Section IX, Chap. I. NOTES. NOTE TO LEMMA I. To establish this theorem, which belongs entirely to the differential calculus, we shall employ the symbolic language, or, as it is sometimes called, the calculus of operations. (See De Morgan's Diff. and Integ. Calc, page 751; also Boole's Diff. Eqs., Chap. XVI.) Let d denote differentiations with respect to k only, and D with respect to 8y only, both k and 8y being regarded as functions of x, and the differentials with re- gard to X being total. Then any order of total differential of any function of k dy may be written {d-\- DY of that function. Now putting v for pair (4), we have v=\{d-\- DY D^ ± {d-\- DY D^lk 8y = id -{• ny D\{d -\- D)"'-^ ± D^'-^kdy = X\{d-^D)^-^ ± D^-^k Sy, (I) where X={d-ir D)D = dD + D\ (2) It must, however, be remembered that ^does not denote a quantity, but merely a mode of differentiation, and that seeming exponents as 2, n, etc., do not indicate powers, but the number of times that a certain mode of differentiation is per- formed. Now from (2) we have Z?2 4-^Z) + ^ = ^_|_X=-(^'^ + 4^), (3) 4 4 4 or 55^ CALCULUS OF VARIATIONS. the first member of (4) denoting differentiation twice according to a predetermined method, the second differential having been rendered perfect by the addition of another differential, just as the square is, in quadratics, by the addition of a square. Hence, solving as in quadratics, we obtain Z) = I|_^± (^2 + 4^)1 I (5) and d^D = l\d±{d'-^^^Xf \^\{d±r\ (6) r dqpoting also a mode of differentiation only. Now put n for m — /. Then in (i), by the use of (5) and (6), we have {d-\- Dr-^ ± n^-i = (^+ ny ±D^ = \ \{d±rY ±{-d± rf \ , (7) the positive or negative sign being used according as n is even or odd. If we first suppose n to be even, and expand both binomials by the binomial theorem, and add the results, then, since each term which does not cancel becomes double, we shall, after multiplying — by 2, have dn _|_ '!kL_l\ dn- 1 ^2 _|_ etc. ^ . (8) Let us next suppose n to be odd. Then the development will assume the same form, because the sign connecting the two binomials will be negative, so that we shall have always (^ + Z))— ^± /)—'::= ^J«'~ + ^^^^^^"- V2-f- etc. I . (9) But since r^ = d^ -^ 4X, we will suppose the values of r\ r\ etc., in (9) to have been found and arranged according to the ascending superscripts of X. Then there will occur in r^, r*, r^, etc., one term involving d, but not X; and this term, when combined with its component outside of r, which will involve d, will always become d» multiplied by some function of n. Therefore the development may be written 4^ (d-^D)'^-^ ± B'^-i = ad^ -{- dd^-'^ X -{- cd^-"^ X^ -}- etc., (10) where a, d, c, etc., are functions of n and numbers merely. Hence from (i) v^e have NOTES. 559 We have now only to abandon the symbols of separation by performing the operations which they indicate, thus: , ■^ dx^ dx^ ^ dx^ and since d-\- D denotes total differentiation, we have ^^5/0; (12) {d^D)^ci8yfJ) = ^^ci8yif). Proceeding in like manner with the other terms in (ii), we shall finally obtain d^ ^? + i where 2,nd ci = — — ak, dx'^ O+i dx'^- hh <:i+i = dx^-^ ck, (13) (14) - ^ i r _L ^(^ - ^) I ^(^ - ^){n - 2){n - 3) , ) 1 ^(^? - I) _L 8 ^^(^ - i)(^? - 2){n - 3) ^^^ ) b = c = -^- i i6 2n — 1 ' ^(n — j)(n — 2)(n — 3) + etc. (15) Now the application of (13) is in reality simple. For we see that for any given value of n its number of terms must always be the same as that of equation (9). Hence there will be but one term when « is i, two when w is 2 or 3, three when « is 4 or 5, etc. Moreover, it will be found from (15) that a is always unity, and that 5 = n when n has any value from 2 to 5 inclusive, and that c = 2 when n is 4, and is 5 when « is 5. ^ 560 CALCULUS OF VARIATIONS. NOTE TO LEMMA II. The integration may be effected as follows: Multiply equation (i) by ut, and subtract the product from (2). Then we have « d d^ d d'^ U =u— Ai{ut)' -\- u —J A'i{ut)" 4-etc. — ut— Aiu' — ut-—Aiu" — etc. * ( d™- d™- 1 Now we know that if F and Q be any two quantities, we shall have pQin) := (pQyn) _ ;^(p'0(«-l) _|_ ''^'' ~ 1^ (/'"0(«-2) _ etC. (2) For let d denote differentiations with respect to Q, and D with respect to F. Then we shall have PQ(.n) ^ d^FQ, {FQf^) = {d-{- DYFQ, (P'0(«-i) = {d -\. Z?)«-i DFQ, etc. Now in the cases which we shall consider n will be a not large positive integer, and it will therefore readily appear by trial that d^ = (d-\- BY - n{d-\- I)Y-^D-\- ''^'^~^^ (^+ Z))«-2/)2 _ etc. ± i>. Hence if we select any term of (i), as d™' u-^Am{uti'^'> or ulAm{utY'^')Y^'^, and put u for F, and the other factor for ^"^we obtain, by the use of (2), u [An, («/)W]W = [uAr^ (w^)^]^ _ m [u'Am (z//)('")]('"-i) + ^(^ - ^ iru"Ar. (uf)('-)](^-^) - etc. (3) But («/)(»») = «/(»«)+ W«7(«-l) + ^ ^ «"/(m-2)_|_ etc. (4) NOTES. 561 Substituting this value in (3), each term may expand into a series, each series having the superscript of the term from which it was derived. Now consider any series, and let its superscript be/, so that it must be the m — p -\- \ xn order, and every term must contain the factor u^"^~p^ Am. Now for an individual term, take that in this series whose order is m — q-\-i, or q -\- i when we begin at the last. This term will be of the form where k = u^-'^-p^ Am «("*-?>, and (5) ^^^ _^ ^^(^- I)- • • ■ (/+!) mjm-i) (^4-1) I, 2, 3 .... (w —J>) I, 2, 3 . . . . (?n — q) ' while / and q must be some positive integer, or zero, and m some positive integer. Now Up and q be unequal, there will, supposing/ greater than q, certainly arise in the series whose superscript is ^ a term of the form dod \ dxP i the signs being like or unlike according asp — q is even or odd. ' Hence, by the theorem of the preceding note, all the terms in 2u -— Am {uff^"' in which/ and q are unequal may be transformed, so that by adding those in which p and q are equal, which have already the required form, we may write ^« ^ ^- («^>'"^ = ^'+ ^ ■^^'' + ^ ^^'" + "'"• ^7) But if the dJ£ferentiation indicated in the first member of (7) were performed, it is evident that the terms which would contain / undifferentiated would be ^u {Am «<'"))('») t=Bt=z ^ut {Am «<'"))('»). (8) Hence it appears from (i) that all the terms containing t undifferentiated will disappear from U, and we shall, therefore, have ^=l^-''+;^^^'"+"=- fe' Therefore / d Udx = Bit' -\ B-i t" A- etc. (10) dx 5^2 CALCULUS OF VARIATIONS. NOTE TO ART. 369. Let A, B and C denote the angles made by the normal with x, y and z respec- tively, and let the Greek letter | (xi or x) denote the angle made with the plane of xz by the plane which contains the normal and is parallel to z, and r} (eta or e) c be / tan — or I tan c, so that, e being the Napierian base, we shall have C en = tan — = tan c. 2 Our object is now to change in (10) the independent variables from x and >/ to ^ and 77. We have . ^ • . 2 sin r „ 2 tan c 2ev sm C = sm 2c = 2 sm ^ cos ; and substituting in this the values of cos A, cos B and cos C, and multiplying by cos ir/, we have t Xcos ^-\-V sin q-\- Zi sin ir/ = D cos irj = — C, (8) the Greek letter zeta being used for convenience only. But since (8) represents the equation of any tangent plane, and every point of the required surface lies on one of these planes, the equation of this surface may be written ^ X cos k -\-y sin ^ + zi sin t7j = — C, (q) where C is no longer constant, but must be such a function of ? and 77 that the variable Z> may always have the meaning just assigned. Moreover, we may regard every point of the required surface as lying at the intersection of three tangent planes drawn indefinitely near, so that in the second ^ may become ? 4" ^^. V remaining unchanged ; and in the third 7 may become 7 + d?/, ^ remaining unchanged ; | and 7/ themselves belonging to the first. Hence we have a right to differentiate (9) with regard to c, and 7; separately, treating x, y and z as constants. Performing this operation, we have X sm ^ — y cos ? = — -, z cos tn = —-• (,10) dq ' drj 5^4 CALCULUS OF VARIATIONS, Next assume, for brevity, di] d^^ d^drf ' dr] ^ drf ^ ' We have also, from (9) and the last of equations (10), — {x cos I +/ sin ^) = C + zi sin iv, zi sin irj = i tan ir/ — - • ' (12) We will now differentiate equations (10) and (11), reducing by means of the equations whose first members are the bracketed quantities, and supposing the last to have been obtained first, so as to employ it in reducing the first. Thus we shall have dx cos ^-\-dy sin ^=\x sin ^ —y cos ^] d^ — i sin irf dz dC dC + [z cos tr/] dV — -^d^ — --dr] = — d\zi sin i7J\ =^ — i tan irf (v d^-\- w dff), d% d% dxsin ^ — dyzos ^ = — [^ cos ^+_ysin ^]^| + — d^-\--jr^drj d'K dK = Z,dh + [zi sin irj] ^^ + — ^| + -—- drj = udk-\- vdrf, d'^Z, d^C, dz cos i?/ = [zi sin iy] dr] + -ir~^ dk +-7-Y drj — v dc, -\- v) drj. (13) d^d^ ^ ' di Now if in the first two of these equations we first make dy zero in each and divide by dx, and then dx zero in each and divide by dy, we shall obtain four equations, the first two of which will each contain — and — , and the last two of which will each contain — - and -7^; and these differentials will then become the dy dy partial differentials sought. Then finding the values of these differentials by common algebraic methods, we shall have d'q _ wi tan irj sin ^ — v cos I dx i tan irj {uw -\- v^) dr} _ vi tan irf sin ^ — « cos I dx i tan ir] {uw -\- v^) d^ _ wi tan irf cos | -}" ^ sin g dy ~ i tan irf (uw -f- v^) drf _ vi tan irf cos I + « sin | dy " i tan irf {uw -\- v^) NOTES. 565 If now we substitute these values in equation (7), observing, if we clear frac- tions, not to remove the imaginary quantity i from the denominator, it will easily reduce to «4-Z£/ = 0. (14) This equation is not itself integrable, but we can easily obtain from it a more general expression, which can be integrated. By making d'\ and dr] alternately zero in the last of equations (13), we obtain dz . dz . , ^ V— —^ cos IT}, w ^=—- cos trj, (15) dc, drf where the differentials of z have become partial, being taken with relation to ^ and 77 only, as separate independent variables. Now since (14) holds for every point of the required surface, its differential with relation to | or 77, or both, must be zero also. Let us therefore differentiate with respect to rj only. Then ob- serving that > d . . ^ i^ „ „ ... 1 -\- -J- 1 iSiniTf = 1 A — — = I — sec2 in = — tan^ in = ttamn't tan ?w, we have ^« dZ, , d . . , , . . dK , dK -T- = -7- (i + — ? tan t?/) -f- 1 tan tt] dri dr}^ ' df} " ' ' drf ' d^'^drj = .tan.^(nan.;;-+— ) +^^^ =- +z..tanev dH . dz , , ~ T^F ^°^ ^'^'vUZJ ^^"^ ^^' div d'^z . dz -J— = TT cos t77 —t sin in. drj drf ' drf ' Hence, by adding, we deduce from (14) d'^z , d'^z , r. dz^ drj^ a partial differential of the second order, the complete integral of which is known to be z=M^iV)^F{^-irj), (17) $66 CALCULUS OF VARLATLONS. /and F denoting any functions whatever, real or imaginary. See De Morgan's Diff. and Integ. Calc, pp. 723, 719, putting i for a^^and rj for t. See also Boole's Diff. Eqs., Chap. XV. It is evident that, having differentiated (14), the present integral is more gen- eral than the integral of that equation ; but it includes the equation of the re- quired surface, which, when the forms of /and i^are assigned, must be deduced in the following manner. We have, from the last of equations (10), Z= I z cos 17] dr}. (18) in which, by (17), z will become a known function of ^ and r}. Then we shall obtain t, by integrating with respect to 7} only, observing to add to the result an arbitrary function of |, which function must be then so determined as to satisfy (14), otherwise the value of C will be too general. Now substituting this final value of C in equations (9) and (10), and then eliminating ^ and 7], we shall obtain the particular equation sought. As an example, assume Whence, by (17) and (18), ^ = I {a^-\- brf) cos ir} drj = — {ac, -}- brj) i sin it} — b cos ir] -f- X, X being any function of I ; its first and second differential coefficients with re- spect to I being X' and X" . Now using this value of C in equations (11), we find easily u:=^ — bzas^iri -\- X" -\- X, w =.b cos ir}, and ti-\-w = X"-\-X=o. The integral of this equation is, as the reader can easily verify by dififerentiation, X =z c cos ^-|-^' sin ^, and by the first of equations (4) we shall have r = b cos — =: -\^-\-e f> b 2\ which is evidently the equation of the surface generated by the revolution of a catenary about the axis of z, that axis coinciding with the directrix ; which is the same result as has been previously obtained. Making b zero while a is not, we have ^ It . Z TC . Z G3— |=— , 1 = -, C£>= — + -, 2 a 2. a the equation of a helicoid. When a and b are any constants whatever, we can still eliminate ^ and r] from (21). As the result merely is given by Moigno, we will here indicate the work without explanation. We have 568 CALCULUS OF VARIATIONS, r^ = b^ cos^ 27] — a^ sin^ irf = r^ (sin^ it] -\- cos^ irf), r^ + a2 = (^2 4. ^2).cos2 irj^ r''-b^z=- («2 _|_ ^2) gin^ iy. i tan z7? = i/--__i! tan {Go-^) = \i tan ?7, 7 + ^)» COS /t; 4" 2 sin ?77 = (f — n, ^77= — ol. (cos ev + e sin im = — 5/ ■ — =z — a^. Whence we obtain z — aoa = — «tan— * ]by ^+a^\ 4/^qr^ ^^^^ This equation evidently represents a surface generated by a helicoid move- ment about the axis of z, of a plane curve whose equation will be given by (21) if in it we make go zero, and r equal to x. NOTE TO ART. 372. When, however, m = — 2, equation (7) will not give the true solution. But by integrating for this case separately, just as before, we shall obtain 2J3 ' THE UNIVERSITY OF CAUFORNIA UBRARY