ACADEMY OF NATURAL SCIENCES OF PHILADELPHIA. Conveyed in 1892 from the estate of JOHN WARNER who died July 16, 1873. NOT TO BE LOANED. Ji. in /vi-'V i^'' ^O. HISTORY OF THK CALCULUS OF VARIATIONS. (JEambriifflc: PRINTED BY C. J. CLAY, M.A. AT THE UNIVERSITY PRESS. ^i;^^^ /l^'/^uy-T-t'^^y^ A HISTOEY OF THE PROGRESS OF THE CALCULUS OF VARIATIONS DURING THE NINETEENTH CENTURY. mmm mtftoi ttBKART CHESTNUT HILL» MASS. By I. TODHUNTER, M.A. FEL]y)W AND PRINCIPAL MATHEMATICAL LECTURER OF ST JOHN'S COLLEGE, . CAMBRIDGE. n^^^^ MACMILLAN AND CO. (2Dambrttrg£: AND 23, HENRIETTA STREET, COVENT GARDEN, HonUon. 1861. \\ PREFACE. In 1810 a work was published in Cambridge under the follow- ing title — A Ti-eatt'se on Isoperimetrical Problems and the Calculus of Variations. By Rolert Woodhouse, A.M., F.R.8., Fellow of Cams College, Cambridge. This work details the history of the Calculus of Variations from its origin until the close of the eighteenth century, and has obtained a high reputation for accuracy and clearness. During the present century some of the most eminent mathematicians have endeavoured to enlarge the boundaries of the subject, and it seemed probable that a survey of what had been accomplished would not be destitute of interest and value. Accord- ingly the present work has been undertaken, and a short account will now be given of its plan. As the early history of the Calculus of Variations had been already so ably written, it was unnecessary to go over it again ; but it seemed convenient to commence with a short account of two works of Lagrange and a work of Lacroi!$:, because they vi PEEFACE. exhibit the state of the subject at the close of the eighteenth century ; the first 'chapter is therefore devoted to these works of Lagrange and Lacroix. The notice of the two works of Lagrange is very brief, for in fact both of them were accessible to Wood- house, and he has given a good account of all that Lagrange accomplished. The notice of the work of Lacroix is fuller because the second edition of that work had not appeared when Wood- hotise wrote ; it was also necessary to indicate two important mis- takes which occur in Lacroix on account of their influence on the history of the subject ; see Arts. 27 and 39. The second chapter contains an account of the treatises of Dirksen and Ohm. The third chapter contains an account of a remarkable memoir by Gauss, which affords the earliest example of the discussion of a problem involving the variation of a double integral with variable limits of integration. The fourth chapter contains an account of a memoir by Poisson on the Calculus of Variations. The great object of this memoir is to exhibit the variation of a double integral when the limits of integration are variable. The memoir is important in itself, and also from the fact that it may be considered to have led the way for those which were written by Ostrogradsky, Delaunay, Cauchy and Sarrus. The fifth chapter contains an account of a memoir by Ostro- gradsky ; this memoir was suggested by Poisson's, and its object is to exhibit the variation of a multiple integral when the limits of the integration are variable. The Academy of Sciences at Paris proposed for their mathe- matical prize subject for 1842, the Variation of Multiple Integrals. The prize was awarded to a memoir by Sarrus, and honourable mention was made of a memoir by Delaunay. The memoir of Delaunay is analysed in the sixth Chapter, and the memoir of PKEFACE. Vll Sarrus in the eiglitk Chapter; tlie seventh chapter analyses a memoir hj Cauchy, in which the results obtained by Sarrus are • presented under a. slightly different form. Here that part of the present work terminates which treats of the variation of multiple integrals. The nest three chapters treat of another branch of the subject, namely, the criteria which distinguish a maximum from a minimum ; these criteria were exhibited in a remarkable memoir published by Jacobi in 1837, which has given rise to a series of commentaries and developments. The method of Jacobi is founded upon one originally given by Legendre ; accordingly the ninth chapter first explains what Legendre accomplished, and also what was added to his results by another mathematician, Brunacci, and then finishes with a translation of Jacobi' s memoir. The tenth chapter con- tains an account of the commentaries and developments to which Jacobi's memoir gave rise. The eleventh chapter contains some miscellaneous articles which also bear upon Jacobi's memoir. The twelfth chapter contains an account of various memoirs which illustrate special points in the Calculus of Variations. The thirteenth chapter contains an account of three comprehen- sive treatises which discuss the whole subject. The fom'teenth chapter gives a brief notice of all the other treatises on the sub- ject which have come to the writer's knowledge. The fifteenth chapter notices various memoirs which have some slight connection with the subject. The sixteenth chapter notices various memoirs which relate principally to geometry, or differential equations, or mechanics, but the titles of which are suggestive of some relation to the Calculus of Variations. The seventeenth chapter gives the history of the theory of the conditions of integrability. The writer has endeavom-ed to be simple and clear, and he hopes that any student who has mastered the elements of the Vlll PREFACE. subject will be able •without difficulty to understand tlie whole .of the work. It may appear at first sight that great disproportion exists between the spaces devoted to the various treatises and memoirs which are analysed. The writer has not considered solely or chiefly the relative importance of these treatises and memoirs, but also the ease or difficulty of obtaining access to them ; and thus a work of inferior absolute value may sometimes have ob- tained as long a notice as another of higher character when the latter could be procured far more readily than the former. In citing an independent work the title has usually been given in the original language of the work, but in citing a me- moir which forms part of a scientific journal it has generally been considered sufficient to give an English translation of the title. Sometimes a mathematician has been named in the history before an account of his contributions to the subject has been given ; in such a case by the aid of the index of names at the end of the volume it will be easy to find the place which contains the account. Occasionally in the course of the translation of a passage from a foreign memoir the present writer has inserted a remark of his own ; this remark will be known by being enclosed within square brackets. The writer may perhaps be excused for stating that he has found the labour attendant on the production of this work far longer and heavier than he had anticipated. It would have been easy to have examined merely the introductions to the various treatises and memoirs, and thus to have compiled an account of what their re- spective authors proposed to eficct ; but the object of the present writer was more extensive. He wished to ascertain distinctly what had been effected, and to form some estimate of the manner in which it had been effected. Accordingly, unless the contrary is distinctly stated, it may be assumed fhat any treatise or memoir PEEFACE. ' IX relating to the Calculus of Variations which is described in the present work has undergone thorough examination and study. This remark does not, however, apply to all the productions which are noticed in the last two chapters of this work. It will he found that in the course of the history numerous remarks, criticisms, and corrections are suggested relative to the various treatises and memoirs which are analysed. The writer trusts that it will not he supposed that he undervalues the labours of the eminent mathematicians in whose works he ventures occa- sionally to indicate inaccuracies or imperfections, but that his aim has been to remove difficulties which might perplex a student. In the course of his studies the writer frequently found that remarks which he intended to offer on various points had been already made by some author not usually consulted ; for example, the considera- tions introduced in Art. 366 occurred to him at the commencement of his studies, and it was not until long afterwards that he found he had been anticipated by Legendre ; see Art, 202. The writer will not conceal his own opinion of the value of a history of any department of science when that history is presented with accuracy and completeness. It is of importance that those who wish to improve or extend any subject should be able to ascer- tain what results have already been obtained, and thus reserve their strength for difficulties which have not yet been overcome; and those who merely desire to ascertain the present state of a subject without any purpose of original investigation will often find that the study of the past history of that subject assists them materially in obtaining a sound and extensive knowledge of the position to which it has attained. How far the present work deserves attention must be left to competent judges to decide; should they consider that the objects proposed have been in some degree secured, the writer will be encouraged hereafter to undertake a similar survey of some other department of science. X PREFACE. The Vriter will receive most thankfully any suggestion or cor- rection relating to the present work with which he may Ibe favoured, and especially any information respecting those memoirs and trea- tises which may have escaped his observation, and those of which he has only been able to record the titles ; see Arts. 394 and 420. The writer takes this opportunity of returning his thanks to the Syndics of the University Press for their liberal contribution to the expenses of printing the work. St John's College, April 15, 1861. CONTENTS. PAGE CHAPTER I. Lagrange. Lacroix 1 CHAPTER II. DiRKSBN. Ohm . 28 CHAPTER III. Gauss 38 CHAPTER IV. POISSON ,53 CHAPTER Y. OSTROGRADSKT .111 CHAPTER VI. Delaunay 140 CHAPTER VII. Sabrus . 182 CHAPTER VIII. Catjchy 210 CHAPTER IX. Legendre, Brfnacci, Jacobi 229 xii CONTENTS. PAGE CHAPTER X. COMMENTATOKS ON JacOBI 254 CHAPTEE XI. On Jacobi's Memoir 311 CHAPTER XII. MiSCELIiANEOUS MEMOIRS 333 CHAPTER XIII. Systematic Treatises • 373 CHAPTER XIV. Minor Treatises 436 CHAPTER XY. Miscellaneous Articles 470 CHAPTER XVI. Miscellaneous Articles 484 CHAPTER XVII. Conditions of Integrability 505 i CALCULUS OF VARIATIONS. CHAPTER I. LAGRANGE. LACROIX. 1. It is the object of the present work to trace the progress of the Calculus of Variations during the nineteenth centurj. It will be convenient to begin with an account of three works which ex- hibit the state of the subject at the close of the eighteenth cen- tury. We shall accordingly in this chapter give an analysis of the treatises on the Calculus of Variations contained in Lagrange's TMorie des Fonctions AnaJytiqiies, in the Lecons sur le Calcul des Fonctions of the same author, and in the Traite div Calcul Diffe- rentiel et du Calcul Integral of Lacroix. 2. The first edition of Lagrange's TMorie, des Fonctio7is Ana- lytiques appeared in 1797, and the second in 1813; the ^vork was also reprinted in 1847. The portion which treats of the Calculus of Variations remains as it was in the original edition, where it extends over pages 198 — 220; we proceed to give an account of this portion. 3. Having treated of ordinary maxima and minima problems in the preceding pages of his work, LagTauge states that the same principles may be applied to determine cm-ves which possess at every point some assigned maximum or minimum property. For example, required the curve at every point of which \y+{in-x)y'\{y + {n-x)y'\ is a maximum or minimum, where y' denotes -—- . ^ dx 2 L ARRANGE. Here it is supposed that at any point of tlie curve y is suscep- tible of variation while x and y are not susceptible of variation ; then according to the ordinary principles of maxima and minima problems we differentiate the proposed expression with respect to y as variable, and equate the differential coefficient to zero. This gives irn-x) {y^{n-x)y] + {n-x) \y^r{m-x)y} =0...(1); therefore , (2fl? — TYi — n)y ^ 2 (m — x) {n—x)' divide by y and integrate, thus we obtain y^ = h{m—x) {n-x) (2), where h is an arbitrary constant. Differentiate the left-hand member of (1) with respect to y' ; this gives 2 {m — x) [n — x) ; hence we conclude that at every point of the curve determined by (2) the proposed expression {y+im- x) y'\{y-h{n- x) y'} is a maximum or minimum according as (m —x) {n — x) is negative or positive. From (2) it appears that the curve is an ellipse if h be negative, and then (m - x) {n — x) must be negative and there is a maximurn ; also the curve is an hyperbola if h be positive, and then [m — x) {n — x) must be positive and there is a minimum. This is the first appearance of a problem of this kind. La- grange intimates that such problems may be proposed involving other differential coefficients besides the first. 4. Lagrange next considers the more common problem of the Calculus of Variations, namely that in which we require the maxi- mum or minimum value of the integral of a functiony {x,y,y',y",...). He uses m to denote what is called the variation of y, and which is usually denoted by %. He arrives at the well-known relation which must be satisfied in order that the proposed integral may be a maxi- mum or minimum; this relation he expresses in the following manner ; f\y) - [/'(3/')]'+ U'iy'W - [/ {y")T + ••• = o. LAGRANGE. *3 He also obtains the ordinary result for the terms which are free fr-om the integral sign, which must likewise vanish in order that the proposed integral may be a maximum or minimum. 5. Lagrange now proceeds to the discrimination of a maximum from a minimum value ; he takes the case in which the function under the integral sign contains no differential coefficient of y higlier than the first. We will here indicate his method, but we shall use the ordinary notation instead of Lagrange's. Let ^ denote ■^ , and suppose /(a?, ^, y) to represent any function the integral of which taken between certain fixed limits is to have a maximum or minimum value. Change y into y + hy and jp into /? + 8p ; thus f{x, y, p) will become where the &c. stands for terms of the third and higher orders in Sy and Bp. Now by means of the relation between x and y given by df fdj dy \dpj ^ ' ' and the fact that the integration is taken between fixed limits, the integral denoted by vanishes. We must then examine the integral if this taken between the fixed limits is negative for all indefinitely small values of hy and Sp, the proposed integral is a maximum when y has the value which satisfies (1) ; if it be positive for such values of hy and hp the proposed integral is a minimum. 1—2 4 LAGRANGE. The integral which we have to examine may be put under the form where X is any fanction of x ; for we can shew immediately by dif- ferentiation that the latter expression coincides with the integral which we have to examine. Now assume \ such that [d^~^V ^ w~^d^)~$' ^^^' then the last expression under the integral sign becomes a perfect square, and the integral may be written ,dY dH _ where ^-rh = t^ - 2^- djp dy dp Thus we have to examine the sign of where x^ and x^ denote the limits of the integration, and \ and X^ are the values of X and By^ and 82/^ the values of By at the respective limits. Let us suppose that By^ and By^ are zero, then we have remaining 1 f^idj- dH Hence we may conclude that if -7^ be always positive between the limiting values of x the proposed integral has a minimum value ; dH and if -^ be always negative between the limiting values of x the wp » proposed integral has a maximum value. Lagrange remarks that this result had been published in the Memoirs of the Academy of Sciences, in 1786 [by Legendre] ; but he adds, that in order to ensure the correctness of the result it ought to be shewn that the value of X determined by (2) does not become LAGEANGE. 5 infinite between the limits of integration, and it is generally im- possible to do this, because the value of A, cannot actually be found. 6. Lagrange takes for example the case in which f{^> y^ p)=p'+ ^mpy ■\-ny''; here -^ is necessarily positive, but Lagrange shews that when n is negative we are not certain of the existence of a minimum. 7. Lagrange then indicates the method to be pursued in dis- criminating a maximum from a minimum when the expression which is to be integrated involves differential coefficients of a higher order than the first. 8. Then leaving the question of the discrimination of maxima and minima values, Lagrange returns to the consideration of the con- ditions which are common to both maxima and minima values. He makes some remarks on the case in which the limiting values of the quantities?/, y', y", ... are not given, but only one or more equations connecting them. He then proceeds to suppose that the function under the integral sign contains, besides y and its differential co- efficients with respect to x, another variable z and its differential coefficients with respect to x. When y and z are independent he arrives at the two well-known relations which must be satisfied in order that the proposed integral may be a maximum or minimum, namely the relation already given in Art. 4, and another which may be obtained from that by changing y into z. Lagrange also gives the results for the case in which y and z and their differential coeffi- cients with respect to x are connected either by a given equation or by the circumstance that an assigned integral expression involving them is to have a constant value. 9. As an example of the theory Lagrange considers the pro- blem of the brachistochrone when a particle moves from one given point to another. Take the axis of x vertically downwards, and let V2^ iji + x) be the velocity which the falling particle has when at the depth x below the origin; then the expression which is to be rendered a minimum is f sj{\+y^ + z'^)dx I s/{h + x) 6 LAGRANGE. where v' = -^ , and s' = -v- ; here we have not assumed that the re- ^ ax ax quired curve is 2^ plane cui've. Hence in order that the integral may- be a maximum or minimum we must have, by the relations referred to in Art. 8, Integrate these equations ; thus y and VIA + x) V(i +y'" + «") V(A + x) V(i +y" + z') are both constants ; hence by dividing the first of these expressions v' by the other we have ^ a constant, and this shews that the curve must be a plane curve. Then by completing the investigation in the usual manner we obtain a cycloid for the required curve. We now proceed to examine whether the proposed integral is thus ren- dered a maximum or minimum. The terms of the second order are (see Art. 5) J 2V(>^ + ^) (1+/ + ^')^ where p stands for ~- and g- for -j- . The above expression may be written 2V(A + a;)(l+/ + 2')^ and as this is essentially positive the proposed integral is rendered a minimum; and thus the cycloid fulfils the conditions of the problem. 10. Lagrange then gives some investigations relating to the conditions of integr ability of functions; tliis is a subject to which a separate chapter will be devoted in the present work. 11. The treatise on the Calculus 5f Variations contained in the Theorie des Fonctions Analytiques is very clear, and although the LAGEANGE. 7 notation is not so expressive as that which Lagrange originally in- troduced, it is far preferable to that employed in the Lecons sur le Calcul cJes Fonctions. We now proceed to give an acconnt of that part of the latter work which is connected with our subject. 12. In the list of Lagrange's works which is appended to the Mecanique Analytique it is stated that the first edition of the Legons sur le Calcul des Fonctions appeared in 1801 as a portion of the second edition of the Seances de VEcole Normale; the Lecons were also included in the 12th part of the Journal de VEcole Poly- technique in 1804. In 1806 a separate edition of the Legons ap- peared containing two additional legons, and these were also in- serted in the 14th part of the Journal de VEcole Polytechnique in 1808. The two additional legons are devoted to the Calculus of Variations. 13. In the edition of the Legons sur le Calcul des Fonctions which was published in 1806, the part bearing on our subject extends over pages 401 — 501 and forms the last two legons. The first of these two legons extends over pages 401 — 440 ; it treats of the integrability of functions, and also contains a sketch of the early history of the Calculus of Variations ; as we do not consider the early history of the Calculus of Variations in the present work, and as we reserve the subject of the integrability of functions for a future chapter, -we shall not here give any account of this part of Lagrange's work. Lagrange states that the work of Euler, entitled MetJiodus inveniendi lineas curvas... would have left nothing to be desired respecting curves which are required to have a maxi- mum or minimum property, if it had been based on an analysis more conformable to the spirit of the Differential Calculus ; La- grange then adds that the object had been attained by his own method given in the Memoirs of the Turin Academy. This method is the well-known use of the symbol S to express a variation. Lagrange states that this method has been explained in most works on the Differential Calculus which have appeared since it was published, and therefore it will be sufficient for him to give merely^ an account of the principles of it ; and accordingly a brief sketch is supplied. 8 LAGRANGE. 14. Lagrange "begins the next legon thus ; — " The method of variations based on the use and combination of the symbols d and 8, which denote different differentiations, left nothing to be desired ; but this method having, like the Differential Calculus, the method of indefinitely small quantities for its base, it was necessary to present it under another point of view in order to connect it with the Calculus of Functions ; I have already done this in the Theorie des Fonctions, but I propose to return to the subject now in order to treat it in a manner more direct and more complete." 15. Lagrange proceeds accordingly to expound the subject with the aid of a new notation. Suppose 7/ = [x, i) expanded in powers of * by Mac- laurin's Theorem. The result is expressed thus, so that dots over the symbol y indicate differential coefficients of y with respect to i, it being supposed that i is made zero after the differentiations. The terms of the series after the first constitute in fact the variation of y ; in this work however Lagrange confines himself to an investigation of the conditions which are common to maximum and minimum values, so that in fact the terms which involve powers of i beyond the first are not used by him. Since the way in which i enters into ^ [x, i) is quite arbitrary it follows that y may have any value we please. 16. Lagrange then arrives at the ordinary conditions for the maximum or minimum value oi JVdx, where Fis supposed to con- tain X and y, and the differential coefficients of y with respect to x. In his investigation he first supposes that x itself does not receive any variation, and afterwards finds the change in his formulae occasioned by varying x. He then proceeds to the case in which V contains besides y another dependent variable z, and its differential coefficients with respect to x ; and he gives the relations which must hold in order that JVdx may be a maximum or minimum both when y and z are unconnected and when they are connected by an equation. LAGEANGE. 9 17. Lagrange gives some investigations relative to tlie maxi- mum or minimum value of a function of two independent variables which involves a double integral. We will indicate how far he proceeds with this problem ; but we shall use the ordinary notation instead of Lagrange's. Suppose V a function of x, 7/, z, p, q, r, s, , dz dz d^z d^z d^z , t,... where ^ = ^,^ = ^, r = ^^, . = ^-^, ^ = ^,, ...; and let U= fJVdydx; then BU= ffBVdi/dx, .,^ dV^ dV^ dV^ dV^ and by = -Y-bz + -T-cp+-^bq + -^- br + ... ds dp ^ dq -^ dr say = Uz + M8p + NBq + PBr +QBs + BSt+... Now by the Differential Calculus and so on ; thus we obtain SF-fZ- — -— — ^ — ]S- \ dx dy dx^ dxdy dy^ / d f T,r<> T^dSz dP ^ ^dSz \ + y- i3IBz + P-=---^Sz + Q-j- dx V dx dx dy j + -rr Nbz + It —^ 7- bz j^ bz . dy \ dy dy dx J In order that SC/'may vanish it is necessary that the coefficient of tz in the first line of the expression for S V should vanish ; that is, we must have T_dM_dN d^ d'^Q d'R ^^ dx dy dx^ dxdy dy^ Then 8 JJ consists of terms which involve only one sign of integra- tion, namely, that with respect to x or that with respect to y. 10 LAGRANGE. Thus Lagrange is correct as far as he has carried the investiga- tion ; but as we shall see hereafter the great difficulty of the ques- tion consists in reducing the terms which involve only one sign of integration to their simplest form, so as to deduce the equations which must hold for the limiting values of the integrals. The difficulty was first overcome by Poisson. Lagrange adds a remark which is not correct ; he says — " The simplest case is that in which the boundary of the surface represented by the equation in x, i/, z is supposed completely given and invariable. Then the variations of z and its dififerential coefficients are zero with respect to the bounding curve and therefore also through the whole extent of the single integrals contained in S Z7, and the condition S 11= is satis- fied of itself." If the bounding curve be given hz vanishes at every point of the bounding curve, but it is not true as Lagrange asserts that S^;, S^'; ••• ^^so vanish. 18. Lagrange illustrates the subject by the discussion of some of the standard problems. He selects the following ; — the shortest line in free space or on a given surface, the brachistochrone in a resisting medium, the curve down which a particle must fall in a resisting medium in order to acquire a maximum velocity, and the surface of minimum area. The first three of these problems had been originally discussed by Euler, the last had been originally discussed by Lagrange himself in the Turin Memoirs. 19. Tlie treatise on the Calculus of Variations contained in the Lecons sur le Galcul des Fonctions is rather difficult, and the nota- tion is extremely uninviting and perplexing. It may be observed that there is a German translation of the two works of Lagrange which we have considered, by'Dr A. L. Crelle ; the translation is accompanied by a running commentary which is incorporated with the text. In the translation of the Lecons the notation of Lagrange is replaced by the ordinary notation of the Differential Calculus. It seems to have been the design of Dr Crelle to translate all the works of Lagrange, but the only work which appeared besides the two we have considered was the Treatise on the Solution of Numeri- cal Equations. . 20. We now proceed to give an account of the chapter on the subject contained in the work of Lacroix. The first edition of the LACROIX. . 11 Traits du Calcul DiffSrentiel et du Calcul Integral appears to have been pulblished in 1797. The second edition of the second volume is dated 1814 ; it contains a chapter on the Calculus of Variations ex- tending over pages 721 — 816. There are some additions and cor- rections extending over pages 716 — 721 of the third volume, which is dated 1819. 21. In his preface Lacroix states that the Calculus of Vari- ations is treated at much greater length than it had been in the first edition of the work ; he considers that he had to effect two things, namely on the one hand to exhibit the Calculus of Variations in all the extent it had reached and with the symmetry which it had gained by means of its peculiar notation, and on the other hand to explain the connexion of the subject with the ordinary principles of the Differential Calculus. He adds that those readers who wish to confine themselves to the Calculus of Variations stiictly so called may begin at page 755. 22. The guide whom Lacroix has principally followed is Euler ; the third volume of Euler's treatise on the Integral Cal- culus contains an appendix on the Calculus of Variations, and in the fourth volume of the treatise a memoir on this subject is given which is reprinted from the Transactions of the Academy of* St Petersburg [Novi Comment. Acad. Petrop. xvi.). Lacroix devotes the first part of his chapter, extending over pages 721 — 754, to an exposition of the method given by Euler in the memoir just cited ; the method is the same as that which was afterwards used b}^ Lagrange in the Lecons sur le Calcul des Fonctions. Suppose y any function of x, say y = ^{x) ; let there be a new variable t, and let {x, t) be any function of x and t which reduces to <^ {x) when t = 0. Then by Maclaurin's Theorem where t is supposed to be put equal to zero in the differential coefficients with respect to t after differentiation. Then (x, t) —

{x), Pfju = f{x), AU = x + zt{x)= X, AP' = x + dx = x', ATI' = x -\-'^ [x) = X ' ; it will follow that f^ Uv = ylr{X), P'M'=7/'=^(l>{x'), nV = i/r(X'), Uv-PM=S2/ = ^jr{X)-(}>{x), U'v' -P'M' = Sij' = ylr {X') -{x'), and 83/' -hy = dhj = ^lr {X') - <}> {x) - i/r (X) + <|) {x). On the other hand, since «% = {x), My = 8^ {x') - B (x) =f{X') - {x)}='>^{X')-^{x)-{y\r{X)-4>{x)}, LACROIX. 19 and 8di/=8{cli{x')-<}>{x)\ =y{r{X') - yjriX) ~ {<}>{x') -{x)} ; therefore dB^ = Sdy. The proof is certainly sound ; it must be noticed however that it is assumed that B always means a change of x into X and doi dx dx T/Til \ d^O) +^(^- )s? + -(^«'-'^' + )S + (^^'- )^' + , rrTUT' diP' d'lQ' d'lB' , J ,^, Thus far there is no difficulty, but Lacroix adds ^ in Article 871; let A denote the total value of /, that is, of jLdx taken between limits determined by the nature of the question ; since this value is a constant it may be introduced under the signs of differentiation and integration, and thus the formula will become hfVdx=Vhx + {{P+AF-IP')--^{Q +AQ'-IQ') + ..\ ; then it will be shewn presently that we shall obtain By' = y"Bx + 0)', By" = y"Bx 4 G>", poissoN. 57 and generally gy«> = y»+«Sa; + «<"> (2). Hence BV= {M-\-Ny' + Py" + Qy'" + ...)hx The coefficient of hx is the differential coefficient of V with respect to x considering y, y', y'\ ... as functions of x-, we denote this by v. And we have dV _ j^,dx du dit ' Thus equation (1) hecomes Juo\ciu du J + j"\Nco+Pco' + Qco"+...)~du. ,-r dV ^ -rrdSx d .VSx Now -T- Bx + F-y- = du du du J Mo and I -A — c?w = V^Sx^^ — VJSx^ . If then we transform to the variable x the second integral con- tained in BU, we shall have BU= V,Bx^ - V,Bx^ + ("'{Nay +F(o' + Qo>"+...) dx. J Xq By the process of integration by parts we can remove the dif- ferential coefficients co', to", ... from under the integral sign. For we have I Fw'dx == PjOj — PgWg — I P'codx, I Q(o"dx = ^i&)/ - Q^(o^ - Q^co^ + Q^'wq + I Q"(odx, and so on. 58 , poissoN. Thus we conclude finally that BU=T+r'Ea>dx (3), where E:^N-P' + Q"-R"' + .., + {p,- q: + b:' -...) CO,- {p,-q:+r: -...).>, + (^,-i^; + ...)<-(a-i?o' + -X + (A-...)<'-(i2o-.-.)<' + 87. It only remains to demonstrate equation (2). We have di , _dy _du dx dx' du Put £c + Sa? and y + ^y in place of x and y in this fraction, subtract the original value y and neglect indefinitely small quan- tities of the second order ; thus dhy dy dSx ^ , _ du du du ^ ~ dx /dxV ' du \duj But, by hypothesis, By = y'Bx + a> ; difierentiate with respect to u, thus du du Hence we have dhy _ dy ^ .d(o , dBx ^11 rini ritt ^ rlnM dy ^ Jft) / , dx dy\ dBx , du du , \" du du) du ^ rinn ri rv% dx dx '/dx\^ du du \duj and ^ POISSON. ^y' du _ dx Jy dx =y'. du day du / dx = 0>, du ,dx yTu= dy ''du.'' 59 thus the value of Sy' reduces to hy' = y"Bx + (o. Starting from this result and from the equation dy' y dx du dx du we shall obtain in like manner By" = y"'Bx + ft)"; continuing thus we shall establish equation (2) for any index n. In equation (3) we may replace co which is under the integral sign by its value (o = Sy — y'Sx, and in the terms outside the integral sign we may replace (Bq, ft)^, co^', (o\, ... by their values «o = ^^0 - ^o'K ' % = ^yo - yJ' K » • • • «i = %i - y/H » «i' = ^yi - yx'^^1 >'" Thus the variation of the integral U will be expressed ex- plicitly in terms of the variations of x and y, and of the variations of the extreme values of x, y, y', y", ... up to the differential coeffi- cient of the order next below the highest which is contained in V, 88. We have thus given Poisson's method of establishing the fundamental formula of the Calculus of Variations in the case of a simple integral. 60 POISSON. Poisson next shews how this fundamental formula may also be obtained by decomposing the integral ZJinto its indefinitely small elements. This is in fact the old method which was used be- fore the invention of the Calculus of Variations, and it is ex- pounded in Euler's Methodus inveniendi.... Poisson however extends the old investigation so as to include the terms relative to the limits of the integral; this according to Poisson had not been done before. 89. We thus arrive at the end of the fourth section of the memoir. In his fifth section Poisson shews how his results are applied to find the maximum or minimum value of the integral U. He says he will not consider in this memoir the question of the distinction of a maximum from a minimum value. He then makes some remarks on the number of constants which will appear in the solution of the difiierential equation furnished by the condition of maximum or minimum, and the manner of determining these constants. He draws attention to the obvious fact that the difiier- ential equation may be immediately integrated, once if iV= 0, twice if iV= and P= 0, and so on. He states that a first integral of the equation can also be obtained when the independent variable does not occm' explicitly in V; because then if we consider £C as a function of?/, this case is analogous to that in which N=0. He shews however that this integral may be found without changing the independent variable in the following manner. We have dV= Mdx + Ndy + Pdy' + Qdy" + Bdy" + . . . ; here the first term Mdx by hypothesis vanishes ; eliminate Ndy by means of the equation j\r-p'+(2"-i2"' + ...=o, thus dV=Pdy' + P'dy+Qdy"- Q"dy + Rdy'" -vR"'dy + ... But the following are identically true : Pdy' + P'dy = d.Py\ .Qdy"-Q"dy = d{Qy"-Q'y\ Rdy'" + R '"dy = d {Ry" '- R 'y" + R "y) POISSON. 61 Thus dV=d.Py' + d{Qy" - Q'y') 'rdiRy'" -R'y" + R"y') + ... and therefore V=G + Py'+Qy"-Q'y' + Ry"'-R'y"-VR"y'-\-... where C is an arhitrary constant. 90. In his sixth section Poisson says that the prohlem of the maximum or minimum of an integral may be decomposed into two parts which may he considered separately. First we may con- sider that x^, y^, y^, ... and x^, y^, y^, ... are given, and proceed to find the value of y in terms of x and the given quantities which makes U a maximum or minimum. The value of y is then to he found from the differential equation and the arbitrary constants must be determined by means of the given values oi x^, y^, y^', ... x^, y^, y^, ... Substitute this value of y and the consequent values of y', y", ...in V; then integrate Vdx from x = Xq to a? = iCj ; thus we shall obtain the maximum or mini- mum value of U, with respect to the form of the function y, in terms of x^, y^, y^, ... x^, y^, y^, .... We may then seek for the values of these latter quantities which make U a maximum or minimum. If we are able to integrate the differential equation and also to obtain the value of I Vdx, then this second part of the problem J X(j can be treated by the ordinary rules of the Differential Calculus. Poisson then shews that by the application of these rules we obtain the same conditions as are found by the Calculus of Variations when the limits of integration are varied, and consequently those terms are introduced which have been denoted by the symbol V in Article 86. [91. It is necessary to make some remarks on this suggestion of Poisson's about dividing a problem in the Calculus of Variations, into two parts. Suppose we have a problem in the Calculus of Variations, and that for example the differential equation N-P'-\- Q"-...=0 62 POissoN. is the differential equation to a circle. We then according to Poisson's method take the ec[uation to a circle which involves three arbitrary constants, and substituting the value of y in terms of x in y we integrate I Vdx ; then bj ordinary Differential Calculus we investigate the values which must be given to the three arbi- trary constants in order to make the last integral a maximum or minimum. If suitable values cannot be determined we conclude that a curve having the proposed maximum or minimum property cannot be found. But even if suitable values can be found we have no right to conclude that a circle does possess the proposed maximum or minimum property ; because we do not compare a circle with any adjacent curve in the latter part of this method, but only one circle with another circle. To determine whether a circle does possess the proposed maximum or minimum property we must proceed as in Article 5, or in some similar way. In fact Poisson's method will be unobjectionable if we know a 'prioriihsX a curve having the required maximum or minimum property must exist ; but it will not be valid to prove that we have found such a curve when we do not know a 'priori that the curve must exist.] 92. In his seventh section Poisson gives the usual extension of his preceding results to the case in which V contains two de- pendent variables y and z and their differential coefficients with respect to the independent variable x. We will give Poisson's result, because it explains the notation which he continues to use in the next section. Let F denote a function of x, y, z and the differential coefficients of y and z with respect to x ; also let U=r'Vdx, ' then BU=T + (''' ilfiSy-y'Sx) + K{Bz-z'Sx)l dx, where T denotes that part of the variation of U which is free from the integral sign. 93. We now proceed to Poisson's eighth section. In a certain case a relation exists between the quantities IT and K, which may be obtained in the following manner. poissoN. 63 The case is tliat in whicli the variable x does not occur ex- plicitly in F, and when we have moreover F= Wz'\ W being a given function of 7/ and z which contains likewise dy dz dz'' dz ' that is, the quantities I t II I II y ^y -y^ which we will denote by ^', ^", .... We shall have then U= f"' Wz'dx = f ""' Wdz. Now by means of the last expression for U, we may exhibit the variation of U by the formula (3) of Art. 86, putting z and W in place of a; and V, and ^', ^", ... in place of y, y", ... The second term of S C will therefore be of the form G{Si/-t'8z)dz, or, which is the same thing, G {z'hy — y'hz) dx ; J a G being a factor which is independent of hy and Zz. In order that this may coincide with the second term of the value of 8 Z7 in the preceding article we must have H {By - y'Bx) +K{Bz- z'Bx) = G {zBy - y'Bz) . This equation resolves itself into E=Gz', K = -Gy, Hy+Kz' = Q; these results we obtain by equating the coefficients of Bx, By, Bz. The third of these results may also be obtained by eliminating G between the other two, and it expresses the relation between H and K which was to be obtained. 64 poissoN. [94. The preceding article is clear ; in what follows there may be some difficulty. Poisson proceeds thus. In the general case where Fis any function oi x, y, y', y'\ ... z, z z", ... let us suppose X an implicit function of another independent variable u, and let us replace therefore y, y", ... z, z\ ... by / \ U t It I I II I II y xy — y X z xz —zx and Vdx by Vx'du. Let us denote relatively to x, x , x", ... by X the quantity analogous to H and K, then we shall find that these quantities are connected by the identity Xx' + Hy+Kz=0. Reciprocally when a given function of x, x, x", ...y, y', y", ... z, z', z", ... satisfies this equation It will be reducible to the form Vx ', so that without changing Its value we can put x'=\, £c"=0, ... , and regard y and z in the given function as functions of x. It may be remarked here in the first place, that the last sentence, reciprocally when &c. is all that is new. Lagrange had given the other part repeatedly ; he appears to have thought it very important. See Miscel. Taur. Vol. ii. page 183, and Vol. iv. page 177; also Legons sur le Calcul des Fonctions, page 412, and page 456. Lacroix also gave the theorem Calc. Diff. et Int. Vol. ii. page 763. In the next place, there is a little difficulty as to Poisson's notation, so that it is necessary to examine the point in detail. Let V denote a function of x, y, z, and the dificrentlal coefficients of y and z with respect to x. Transform these differential coefficients into differential coefficients with respect to a new inde- pendent variable u, so that V may be transformed into a function of X, y, z and the differential coefficients of these variables with respect to u ; we will denote the transformed function by v. Then put JJ= I Vdx = \V -y-du= \v -r- die. J J du J au We have now two modes of expressing B U; we shall confine our- selves to the unlntegrated part. T^hls may be written thus poissoN. 65 or thus I[a (S. - f ^ Bu) + S (8^ - I S„) + C (8. - 1- Bu)] du. Here Y and Z are olbtained in the ordinary way from F; and doc A, B, C are obtained in a similar way from v -j- . By comparing the two results, remembering that the integration in the first is with respect to a;, and in the second with respect to w, we obtain du ' du ' dx dxj du . dx T^dy ^dz ^ A~^+Bj- + Gj~ =0. du du du The last resnlt will also follow from the first three by eliminat- ing Y and Z. The last result must be what Poisson denotes by Xx + Hi/' + Kz' = ', his notation is objectionable however, because he had previously used H and K for what we denote by I^ and Z. Next let us consider the reciprocal theorem which Poisson enunciates. Let (fy denote any function of a;, x, x", ... y, y', y", ... z, z', z", . . . which satisfies the condition Ax' + B7/'+ Cz' = 0. Transform the independent variable from u to x and let ^/r — be what ^ becomes ; the assertion then is that ^jr will be free from u, that is, i/r will not contain -y- , -r-^ , ••' To prove this we observe that if 'xjr did contain such terms we should have, considering only the unintegrated part of the variation, a result of this form BJ^dx =j\j{Bu - g Bx) .+ r(8y - I Bx) + Z{Bz-p^ Bx)]^ dx. 66 poissoN. But again taking the unintegrated part of the variation BJc}>du = J\a {Sx - ^ 8.0 + 5 (% - ^ Bu) +C{Sz- ^^ SzOJ du. Now by supposition l-yfrdx = lyjr-j-du= l(f)du, and therefore the variations of the two expressions must coincide. But Bu dis- appears from SJ + P(o' + Qco^ + R(o" + S(o; + r« , + ... or, which is the same thing, Z V = V 'hx + Vhj + N(o -vP(o' + Q(o, ^-Ray"+S(o;+T(o,,+ (2). 103. The twentieth section contains some reductions of the variation of a double iutegi'al. Consider the definite integral U= ffVdxdy. By the known rules for the transformation of double integrals, if we consider x and y as functions of two other variables u and v, we must put , , fdx dy dx dy\ , , dxdy = {-=. — Y- — 7- -T-\dud'0', ^ \du dv dv duj so that we have 'U=\\v(^^^-^^f]dudv, J J \au dv dv duJ Now put x^Bx, y + By, z + Bz in place of x, y, z under the in- tegral sign. From what was said above it will be sufficient that Bx, By, Bz should be arbitrary functions of u and v, and it will not be necessary to vary the limits relative to ii and v in order that the integral U may vary in the most general manner both with re- spect to the limits relative to x and y, and with respect to the form of the function z. The complete variation of ZJwill then be ■,U=[(('^^'^-^f)BVdudv J J \du dv dv duJ . 1 fdx dBy dx dBy dy dBx dy dBx\ ^^ , \du dv dv du dv du du dv J 76 POISSON. But by the formulge of the preceding article we have dx dZy dx dhy _ idx dy dx dy\ dZy du dv dv du \du dv dv duj dy ' dy dSx dy dBx _ fdx dy dx dy\ dBx dv du du dv \du dv dv du) dx Hence SZ7= dhx „ dZy'X idx dy dx dy\ -, , », dx dy ) \du dv dv du) ' that is, by restoring the variables x and y In this formula the limits are the same as those of U. Now substitute the value of SF given by equation (2), and observe that T^,- -r^dSx d. VSx V bx+ V -J- = — J , dx dx V,hy+V^^ = ^-4^', ^ dy dy thus SZ7= + "/ VBxdy Vhydx -fjvSxdy VBydx + I (iVft) + Pco' + Q(o^ + Boo" + Scol + Tw,, + ...)dxdy (3). By the method of integration by parts we may remove the differential coefficients of w from under the double integral sign. For I \P(o dx dy = Pft) dy Pci) dy] — j IP'co dx dy, ljQco^dxdy= jQwdx —(JQcodxj—lJQ^codxdy. POISSON. By two successive integrations we have 77 \\Rm"dxdy = 1 1 Tod^^dxdy = By integrating first with pespect to y and then with respect to x we obtain \Ra>'dy J — f Rw'dy j JB'cody + ( Ii'Q)dyj + jjB"(odxdy, [To), dx _J -(JTco^dx) jT;codx + ( T^oidx] + tlT^^wdxdy. \J J J J I iSco^'dxdy = Sco'dx S^wdy — I Sco'dx + (jS^cody) + I j S^'co dx dy j by performing the integrations in the reverse order, we obtain \ \ \ Sco'^ dx dy = Sq)^ dy S'codx — [ I Sco^ dy + ( 1 S'codx ] + 11 S^'co dx dy. For the sake of symmetry we may use the half sum of these equivalent expressions, that is I lSco^'dxdy = - I Sco'dx S^cody Sco^dy S'codx .1(^1 Sco'dx) -ll^jSco^dy + 1 1 S'^codxdy. The subsequent terms in the last part of the formula (3) may be transformed in a similar manner. Thus the expression for S U will become finally BU=V+jJHcodxdy (4), 78 POISSON. where for shortness, H=N-P'-Q, + B " + S;+T,-... T = Vhxdy -l[vhxdy\+ Ivhydx - llvhy dx \P(ody + I Qccdx — jE'oody— - i S^cody — - I S'codx— j Tfiodx-\- . . . — (jPcody + j Qoodx — IR'aody— - I S^cody— - I S'codx— j T^codx+...\ + IIi(o'dy + — iSoo'dx+ jTco^dx + - ISoo^dy — ... - f Bco'dy + - So)'dx+ \Tco^dx + - jSco^dy- ...j + The two expressions which have been found for jjSco^'dxdy must be identically equal ; hence we have Sw dx — I 1 8(o' dx Sco^dy S^cody S'oodx -{■[jS,o>dy^ + fj S'codx j . Sco^dy This may be written j(Sco' + S'(o]dx -(j(Sco' + S'co]dx] jfSco^ + S,(o) dy - (j(S(o, + S^co) dy\ (5). This will be verified presently (see Art. 106). We may ob- serve here that Sco + S'co is the partial differential coefficient of Sec with respect to x before substituting the value of y obtained from one of the limiting equations. But the value of [Sco' +S'oo) dx after we substitute for y its value is no longer a complete differential with respect to x and thus cannpt be integrated immediately. Similar remarks apply to the term {^Sw^ + S^w) dy. [This remark guards against the error indicated in Art. 27.] poissoN. 79 104. The twenty-first section. For C/" to be a maximum or minimum we must have B U= 0. The double integral included in equation (4) cannot be reduced to simple integrals because co is an arbitrary function of x and y ; it will therefore be necessary that the two parts of this formula should separately vanish. Thus we obtain r = o, H=0, for the e(|uations which must be s?ttisfied in order that the double integral which we are considering should have a maximum or minimum value. The second equation will serve to find z in terms of X and y ; this equation will be in general a partial differential equation of the order 2n if Fbe of the order n. The first equation will decompose into others the number and natm*e of which in the difierent cases which may occm- we will investigate in the sub- sequent articles. This is the most delicate part of the question. The preceding analysis may be extended without difficulty to triple and quadruple integrals, &c. In the case of a triple integral, for example, we shall obtain for the variation an expression analogous to that in equation (4) ; this expression will consist of a triple integral, and of another part containing only double integrals which relate to the limits of the triple integral we are considering. We might also suppose that the quantity under the triple integral sign involves unknown functions of the independent variables, and that these functions are independent, or that they are connected by given partial differential equations. We shall not stop to consider these questions, since they present no new difficulties and no useful applications. The determination of the relative maxima or minima of mul- tiple integrals can be reduced to the determination of absolute maxima or minima by the method of the thirteenth section, which is obviously applicable whatever may be the number of the inde- pendent variables. Thus, for example, if the first of the double integrals 1 1 Vdx dy, 1 1 Tdx dy, j / Wdx dy, ... is to be a maximum or minimum, and at the same time the others 80 POISSON. are to have given values, the problem amounts to investigating the absolute maximum or minimum value of [{V+aT+hW+&c.)dxd^; where a, h, ... are unknown constants which must be determined bj means of the given values of the integrals. We suppose here that these integrals and the first integral are all taken between the same limits. 105. The twenty-second section. [The results from this point to the end of the memoir were not known before the publication of the memoir.] Let us now examine the equations relative to the limits of U which are necessary for the maximum or minimum of this double integral, and which must be deduced from the condition F = 0. In order to render the reasoning easier to follow, we will sup- pose that X, y, z are the rectangular co-ordinates of any point of the surface determined by H=0, and that the integral U corresponds to a zone of this surface comprised between two closed curves which will be given or which will have to be determined. Let ABC be the projection of the exterior curve upon the plane of {x, y), and DEF that of the interior curve upon the same plane (see fig. 2). The integral relative to x and y which h ZJ represents will extend to all the elements dxdy of the plane area intercepted be- tween these two curves. It may however also be considered to represent the excess of a double integral extended to all the ele- ments of area enclosed by the curve ABC over the same double integral extended to all the elements of area enclosed by the curve DEF. Now hU reduces to V since by supposition 11=0; and P^p(i)_p(o) ^i;^ere T*'' denotes that part of SU which arises from the area bounded by ABC and F'"' that which arises from the area bounded by DEF. Since these two limits ABC and DEF are in general independent of each other, the equation F = will decompose into two others, namely. It will be sufficient to consider one of these ; the other will be of the same form and susceptible of the same transformations. POISSON. 81 We had in the twentieth section the equation \\~^dxdy= jvSi/dx -(jVBi/dx). [It has been already intimated in a remark on Art. 101, that Poisson does not use his symbols in the sense which he assigned to them ; the terms in square brackets refer to the upper portions of a curve, and those in parentheses refer to the lower portions of the same curve.] If this double integral relates to the area ABC, the integration relative to i/ which has been effected, is to be extended from one to the other of the two ordinates PM and FM', which correspond to the same abscissa x. We will suppose that it is extended from the smaller ordinate PM' to the gTcater P3I; that is, we consider the variable y to increase and so dy to be positive. As the element of area dxdy is essentially positive, it follows that dx must be regarded as positive in the two simple integrals which are indicated. The first will correspond to the part AMB of the curve ABC and the second to the part AM'B supposing that A and B are the two points of the curve where the tangents are parallel to the axis of y. Let s denote the length of an arc of the curve ABC measured from any fixed point of the curve up to the point M, and let I be the com- plete perimeter of the curve. Then we shall consider s to increase from s = to s = ?, and thus the differential ds to be positive. Let /S be the angle comprised between the exterior normal MN and the produced part of the ordinate PM. Since dx is the projection of ds on the axis of x, we shall have dx = ± cos /3 ds ; the upper or lower sign must be taken according as cos ^ is positive or negative. But the angle /3 is acute in all the part AMB of the cm've ABC and obtuse in all the part AM'B; hence we shall have dx = cos ^ds throughout the extent of the integral VBydx and dx = — cos /8 ds throughout the extent of the integral ( 1 VBy dx Hence we conclude that their difference will reduce to a single integral relative to s which will extend throughout the whole curve ; that is, we shall have 6 82 POISSON. I Vhydx — { \ VBy dxj = j Vcos^By ds. Similarly we shall have Vhxdy — i I VBxdy j = I Fcos a hx ds, where a denotes the angle which the exterior normal MN makes with the produced part of the abscissa of the point M. By similar reasoning we maj reduce to a single integral each of the differences of two homologous integrals of which the expression V is composed. Thus the equation F "' = will be transformed into the following : F( cos aBx-\- cos /SByjds i + 1^' \{P-R' -\S. + •••) cos a + ((3- ^, - ^ ^' + ...)cos^ + 1 (i2cosa+ ->S^cosj8— ... ] ft)'c?s + I ( Tcos ^ + - S cos a— ...ja^ds ids + ...=0. (6). The figm'e supposes that each line parallel to the axis of y meets the closed curve ABC in only two points ; but this trans- formation of the equation T '^' = would still hold if the number of intersections, which must be even, were greater than two ; we should then take successively for the two ordinates P2I and PM' which correspond to the same abscissa, those of the first and second intersection, those of the third and fourth, and so on. 106. In his twenty-third section Poisson verifies equation (5) of the twentieth section. In fact, as we have just seen, the part of this equation which belongs to the exterior curve is the same thing as -' d.Si dx y\ ^ , ['fd.Sa)\ , - 1 cos p as = \ I — J — j cos a as, PoissoN. 83 ■where tlie parentheses denote that each partial differential coefficient is taken with respect to one of the variables x or y before we sub- stitute in Sco the value of the other variable deduced from the equation to the curve ABG. If we differentiate with respect to x after substituting the value of y, we have d . 8(o _ /d. 8(o\ /d . Sco\ dy dx \ dx ) \ dy ) dx^ T . d. Sco d . Sco dx d . Sco ^ and since — ^ — = — ^ 7- = — = — cos 8, ds dx ds dx it follows that n[d.8(o\ „, c'd.Scoj n/d.8co\dy ■ ^, J \ Cljj / J Q U/O J \ ClU J U/JU Now I —^ — ds= Sco + constant ; and since Sco has the same value at the two limits s = and s = l which belong to the same point of the closed curve ABC, it follows that 'H.Sco I Jo ds = 0. ds Hence the equation reduces to Pfd.8ca\ ^, f^/d.Sco\dy ^, By help of this result we see that the equation which we are to verify may be written thus, But if a and h denote the angles which the tangent at any point M. of the curve ABG makes with the axes of x and y, we can take in this equation where the differentials dx and dy may be positive or negative, dx~ cos a ds, dy — cos h ds, 6—2 £( 84 poissoN. Thus the equation is transformed into the following : '^^"j jcosa cosa + cos^ cos4 — =0; dy )\ J cosa this result is obviously true since the factor cos a cos « + cos /3 cos h is the cosine of the angle comprised between the tangent and the normal at the same point M of the curve ABO, and is therefore equal to zero. It is evident this verification applies in the same way to the part of equation (5) which belongs to the interior curve. 107. In his twenty-fourth section Poisson effects some trans- foi-mations of the equation (6) of Art. 105. The applications of the preceding formulse to geometry and mechanics relate to problems where the function V depends on the inclination of tangent planes and on the magnitude of radii of curvature. In order then to avoid useless complication, we will suppose that the highest differential coefficient contained in Fis of the second order. In this case the equation H= involves partial differential coefficients of the fourth order, and the first member of equation (6) is reduced to its first four terms. But in order to be able to deduce from this equation (6) the conditions relative to the second limit of Z7, it is necessary to transform its third and fourth terms, and to reduce the three variations w, w', and co^ to two only. All the terms of equation (6) are integrals relative to the arc s of the curve ABC, where s is the independent variable and ds is constant and positive. Under the integral sign z is regarded as a function of x and y, which is obtained from the equation to the re- quired surface, that is, from the complete integral of the equation H= 0. The variables x and y are implicitly supposed to be functions of s determined by the equation, known or unknown, of the curve ABC. Thus by differentiating w with respect to s, we have dw _ , dx dy ds ds ' ds ' hence since dx^ + dy^ = ds\ we may write , _dx d(o f. dy _^dy dw ^ dx ds ds ds ^ ' ds ds ds ' where 6 is an indeterminate variation. poissoN. '85 The differentials dx and dy are as in the preceding article capable of changing sign in the course of the integration, that is, as we proceed from point to point of the curve ABC Since the angles a and /3 relate to the exterior part of the normal MN it is easy to see that we shall have at any point M, dy ^ dx cosa = — r-j cos 6=-^-. as as Substitute these values and those of ^.f-^(^-^)^ J. ^f CL*Xj J, Qj CtOC CtXj Qj Q/ du -^T" •D ^if ^dx dy ™ dx? _ „ ds^ ds ds ds^ [The value of Z agrees with Poisson's ; those of X and Y differ since Poisson omits the last three terms of each,] Thus equation (6) finally becomes + ! Z0ds = O (7); and this is the simplest form it can take. 108. The twenty-fifth section relates to the case in which some condition is given. In the problems to which this equation can be applied, it will sometimes happen that the length of the exterior curve to which it relates is to have a given value ; or more generally that one or more integrals taken throughout this length are to have given values. It will be sufiicient to consider one of these integrals ; for similar remarks would apply to the others. For greater simplicity we will suppose that the differential function which occurs under the sign of integration is only of the first order. At any point of the exterior curve then, let dx _ , dy _ , poissoN. 87 let W denote a given function of x, y, z, x, y\ and suppose tliat fWdz = a ; a being a given constant, and the integral extending througliout r dz this curve, so that it is the same thing as I W-j-ds. In order to introduce this condition it will be sufficient according to the remark in the thirteenth section to add to Z7 the integral /TFcZs multiplied bj an undetermined constant which we will denote by c. Thus the first member of equation (7) is augmented by the term cSJWdz. Now if we put dW_ • dW dW dW ~dh~^' ~c this term has for its value dx-i"' AT""' S^ = "'' W^"' /[('^-S(^---'«^) + (^-s)(^^-^'«^)' dz considering x and y as functions of z in the formula of the seventh section (Art. 92) and observing that the part outside the integral sign vanishes because the curve which we are considering is a closed curve. Suppose that this curve is to lie on a smface which we wiU denote by the differential equation dz = pdx + qdy, where p and q are given functions of x, y a^id z. We shall see presently (Art. 114) how the case of a curve unrestricted is com- prised in the present. The co-ordinates x, y, z of any point in this curve, and also the varied co-ordinates x-\-hx, y + hy, z + Bz must satisfy the equation to the given surface ; we may therefore differentiate that equation relatively to the characteristic B; thus we shall have Bz =])Bx + cjBy as well as dz =pdx + qdy. Hence Bx — x'Bz z=q(-j^ Bx — f ^^] ? pv (P. fct'X ^ CLy ^ By-yBz=x^[^^By-^^Bx^ 88 poissoN. If then, for abbreviation, we put dm J. dn J the term which is to be added to equation (7) becomes e^ycp-hq)(^^hy-^^h^ds. Thus it has the same form as the first term of this equation ; consequently in order to introduce the condition that the integral / Wdz is to have a constant value, we have only to change in equation (7) Finto V+ c{kp — }iq). The constant e will have to be determined in every case from the value a of the integral fWdz. 109. The twenty-sixth section. Let us now observe the re- sults which may be deduced from equation (7) thus modified if necessary. Let us put -j-Zy — -^Zx = cosaSic + cos/3Sy ds Q) = Sz- z'Bx - zBy = (j}\fl + z^ + zf. The point M of the curve ABG whose co-ordinates are x and y being transferred to the position indicated by the co-ordinates x-'tZx and y + Sy, we see by the value of e that this variation denotes the displacement of M projected on the normal MN. The cosines of the angles which the normal to the required surface at the point {x, y, z) makes with the co-ordinate axes are respectively Thus the variation is the projection on this normal of the displacement of this point (a?, «/, z) when its co-ordinates become x-\-hx^ y + By and z + Bz; and in equation (7) this displacement belongs to any point of the exterior cm^ve. As to the third arbi- trary variation contained in equation (7), namely 6, this depends on the change of inclination experienced by the tangent plane to poissoN. 89 the required surface at any point of the exterior curve. [In fact we have from the equations in which d first occurred . dx ,dy and if we insert the values of w, and &>' from Art. 102 we have = (8^, - z;sx - zjy) -£ - (Bz - z"Bx - z;hy) -£ . Thus 6 involves Sz^ and Bz and is thus connected with the change of inclination of the tangent plane.] Now if the second limit of U is not restrained by any given condition, the three variations e, (j), 6 will he completely arbitrary and independent ; hence in order that equation (7) may subsist it will be necessary that the coefficients of these variations under the integral sign should be separately zero. Thus we shall obtain three equations, ^=»' ^S-^'l = »' ^=« w- When the second limit of U has to satisfy some given con- ditions the three variations e, ^, 6 are no longer independent ; then the equations (8) or at least one or two of them will not hold and must be replaced by others. The following are the principal cases which may arise. [Five cases are considered which will occupy the following five articles, extending to the end of Poisson's twenty- sixth section.] 110. Suppose that the exterior curve is fixed and given, and let us represent its two equations by f{x,y,z)=Q, F{x,rj,z)=0 (9). From the signification of e and ^ it follows that these quantities now vanish; thus the first two terms of equation (7) disappear. The first two equations of (8) will now no longer be necessary and they will be replaced by the equations (9). Let us farther suppose that the required surface is to touch throughout the perimeter of the exterior curve a fixed and given surface, as for example the surface which has for its equation 90 POISSON. F {x, y, z) = 0, and tlie differential equation of which we will repre- sent by dz=pdx + qdi/. It will be necessary on account of tliis contact that ^ = z', q = z^ for every point of the exterior curve. These however are not two new independent equations ; for since the curve is aheady the intersection of the required surface and the given surface, the dif- ferential dz taken along its direction has the same value whether it be obtained from the equation to the first surface or from the equation to the second surface; thus we have already the rela- tion pdx + qdy = z'dx + z^dy ; and by means of this relation one of the equations ^ = s' and q^=-z^ is a consequence of the other. On the other hand the variation ^ and consequently w will be zero, not only for all the points of the exterior curve, but also for all those of an indefinitely narrow zone of which this curve forms part ; we may therefore differentiate the equation to = along the direction of this curve and along any other direction ; thus we shall have throughout the whole perimeter of the curve _ = 0, ft) = 0, ft>, = ; thus the quantity which occurs in the twenty-fourth section vanishes, and the third term of equation (7) vanishes. Thus in this first case the three variations e, (/>, 6 being zero, the equation (7) vanishes ; the equations (8) which were deduced from (7) do not hold, and they must be replaced by the equations (9) which will be given in each particular problem, and by one of the equations p = z, c[ = z^. 111. Suppose that the exterior curve is fixed and given, so that e = and w = 0, and suppose that the second limit of U is not restrained by any other condition. The equation « = can now be differentiated only along the direction of the given curve; we liave then POISSON. 91 the factor 6 remains indeterminate, and we must have Z= in order to satisfy equation (7) which now consists of only its third term. In this second case the third of the equations (8) holds, and the other two are replaced, as in the first case, Iby the two given equa- tions of the exterior curve. 112. If this curve is not fixed but only constrained to lie on a given surface which is determined by the equation F{x,y,z)=0 (10), then the co-ordinates x, y, z and also x + Sa?, y +hy, z +hz must satisfy this equation. We may therefore differentiate with respect to the characteristic S ; thus if we represent the ordinary differential equation by dz = ^dx + qdy, we shall also have simultaneously Bz =pBx + qZy. Hence the variation on will be given by the equation (ti z= (^p — z) hx -{■ {c[ — z) Sy. Suppose moreover that the required surface is to touch the given surface throughout the perimeter of the exterior curve. We shall have the two relations p = z' and q = z^, one of which is a conse- quence of the other, as we have shewn in the first case. These relations will make to vanish for all points in an indefinitely nar- row zone comprising the exterior curve ; hence as in the first case we conclude that ^ = 0. Since the variations co and 6 are zero the equation (7) is reduced to its first term ; and in order that it may hold whatever may be the value of the indeterminate variation e we must have V= ; or rather V+c{Jc2)-hq)=^o!. (11), if we suppose, as before, that the value of a certain integral JWdz is given. Thus in this third case the equations (8) are replaced by the equa- tions (10) and (11) together with one of the two relations p = z' Sindq = z^. 92 POISSON. 113. Suppose that the exterior curve is still constrained to lie on the surface determined bj equation (10), but that the tangent plane to the required surface is not subject to any restriction throughout the perimeter of the exterior curve ; the expression for ft) given in the preceding case will still hold, but there will be no resulting limitation for the quantity 6, which will remain altogether arbitrary and independent of hx and hy ; the coefficient of 6 in equation (7), that is Z, must therefore be zero. Substitute the expression for w in this equation and it will become /;[( /:[(^' 4:-^'l)(^-')-4'' dx ds -s^dy ds te-^,) + ^f Bxds By ds = 0. But as the two variations Bx and By are arbitrary and inde- pendent their coefficients must be separately zero ; if we add then to V the part which arises from supposing the integral JWdz to have a given value we shall obtain But one of these equations is a consequence of the other ; for if we multiply them crosswise and suppress the factor common to the two products we obtain {p-z')dx={z^-q)dy; and this equation, as we have seen in the first case, follows from the fact that the required sm-face and the given surface intersect in the exterior curve which we are considering. These equations may be written in the follownig manner Y{p- z') dx = [X (j)-z')+V-\-c {kp - qh)] dy, X {q - z) dy=[Y{q- z} +V+c {kp - qh)] dx ; multiply these equations and reduce, thus we obtain V+c{kp-q7i)+X{p-z)+Y{q-z;)=0 (12). poissoN. 93 Thus in this fourth case the third of equations (8) holds, and the two others will be replaced by equation (12) together with that of the given surface of which the differential equation is represented by dz =jpdx + qdy. 114. Lastly, suppose that the tangent plane to the required surface may vary arbitrarily throughout the perimeter of the ex- terior curve, and that this curve is not constrained to remain upon a given surface. The equation Z=0 will still hold. We may write equation (12) in the form V+ c [hz' - hz) + (X+ ck) [p - s') + ( Y- ch) {q-z)=0', multiply by dx, and put {z^ — q) dy for [p — z) dx ; thus we have [ F+ c i^z - As; j] dx-\-^ Ydx - Xdy - c {Mx + My)] {q - z) = 0. But the quantity q is altogether arbitrary, since now the sm-face which had for its differential equation dz —pdx + qdy is not given ; the preceding equation must therefore separate into two, and con- necting them with Z= we shall have for the three equations be- longing to this fifth and last case F+ c ijcz - hz) =0, 1 Ydx — Xdy — c{Jkdx-\-hdy) =0, I (13). Z=0. J These equations coincide, as they should do, with the equations (8), when we put c = 0; this amounts to suppressing the condition relative to a given value of the integral JWdz ; so that now there is no longer any given condition by which the second limit of U is restricted. 115. The twenty-seventh section. The reasoning already given applies equally to the first limit of U; and by the details which have just been given we see that the conditions of a maxi- mum or minimum of this double integral consist in this, that for each limit the required surface must satisfy simultaneously three known equations which will either be directly given, or which may be formed for the different cases which can occur in the manner we have explained. These two systems of three equations will serve 94 poissoN. for the determination of the four arbitrary functions involved in the complete integral of the equation H=0. When the differential function V is only of the first order we shall also have E = 0, S=0, r=0; the partial differential equation H=0 will not he of a higher order than the second ; we shall have X=P, Y=Q, Z=0; and the equations of the preceding article will simplify and will reduce to two for each limit of U. If we wish to apply the formulae of the preceding article to the case of a single integral, we must suppose that the qu.antity F is a function only of x, z, z', z" ; hence we shall have ^=0, S=0, T=0. It will he necessary at the same time that the zone of the required surface to which the integral U will belong, should be comprised between two planes parallel to that of {y, z). The curve ABG will then reduce to two straight lines parallel to the axis of y, the limits of two oval curves of which one dimension is indefinitely increased ; and as in the equations with which we are concerned the differentials of x and y relate to this curve and the differential ds is supposed constant, we must put dx = 0, d'^x = 0, dy — ds, d^y = 0. The condition relative to the length will no longer hold, so that we also must suppress the terms which thence arise, that is, put c = 0. Under these circumstances the equations of the twenty- sixth section will coincide in all cases with those which would be derived from the fifth case (Art. 114), observing that the quan- tities which were represented by y and Q in that section are now represented by z and B, and that the function V being supposed of the second order, the quantities R, S, &c. of that section are zero. This coincidence would supply, if that were needful, a confij-mation of our analysis with respect to double integrals. 116. In his twenty-eighth section Poisson makes some remarks on the mode in which the arbitrary functions are to be determined poissoN. 95 in some problems. There are particular problems in wliicli the curve which forms the inner boundary of the required sm*face accord- ing to the hypothesis of the twenty-second section (Art. 105) does not exist, and in which consequently the conditions relative to this curve must be replaced by others, in order that the arbitrary func- tions which are involved in the general integral of the equation H= may not remain undetermined, and that these problems may be completely solved. This circumstance might occur, for example, in the question where we have to find a sm'face of which the area should be a minimum between certain limits. The equation 11=0 is then a partial differential equation of the second order, and its integral involving two arbiti-ary functions is known in a finite form. Now if the minimum area is to be a zone included between two eiven curves, we see that these two curves through which the required siu'face is to pass will theoretically serve to determine the two arbitrary fmictions which occur in the equation to the surface, that is, in the integral of the equation 11= 0, the only difficulty being that which arises from the complicated form of this integral. We see too that these two curves might be exchanged for other pairs of conditions. But if we require that the minimum area should be all that portion of the surface which is bounded by the exterior curve, it seems then that the integral of the equation H=Q will have a greater degree of generality than the problem, and that the given curve will not be sufficient for the determination of the two arbitrary functions. 117. In order to remove this apparent indeterminateness, sup- pose we exchange the rectangular co-ordinates x and y for polar co- ordinates r and 6, where r is the radius vector and 9 the angle which r makes with a fixed line drawn through the origin in the plane of {x, y). Put the origin within the boundary formed by the projec- tion DEF oi the interior curve (Art. 105) when such curve exists. Let r=f{e), z = <^{e), be the two equations of this curve ; and r = F{e), z = ^{e), those of the exterior curve of which ABC is the projection. k 96 POissoN. For the zone of minimum area the values of r will extend from r =f{0) tor= F{d), and those of 6 from 6 = 10 0=2-77, and the arbitrary functions which occur in the integral of -3"= must be determined so that z should become j>{6) and ^{6) for r=f{6) and r = F{6) respectively. Outside the zone, that is, for values oir less than y(^) or greater than ^(^) whatever 6 may be, the ordinate z will be subject to no limitation and can become infinite. But if the minimum area is to be all that portion of the surface the projection of which is bounded by the curve ABC, the values of r will extend from r = to r = F{6) for every value of ^, and throughout this extent the ordinate z must be finite. We shall therefore suppress in this case that portion of the integral oi H=0 which would become infinite when r = ; and the integral thus modified will be reduced to the degree of generality which the problem has ; so that the single condition that z should be equal to ^{6) when r is equal to F{9) will suffice for completing the solution of the problem. Thus the solution of the question of the minimum area and of similar questions, separates into two problems which are quite dis- tinct so far as relates to the determination of the arbitrary fimctions. I only here indicate this distinction which I will take up on another occasion. If the required surface is closed on all sides, so that for example we have to find the surface of greatest area which incloses a given volume, the conditions for this relative maximum will not furnish any equation suitable for determining the two arbitrary functions which the complete integral of the equation H= when applied to this problem will involve. It is by means of other considerations that this integral must be reduced so as to contain only three arbitrary constants, namely the three co-ordinates of the centre of the sphere which solves the problem ; the radius of the sphere will be determined by means of the given volume. I propose to con- sider this particular question in another memoir. [It does not appear that Poissoij ever returned to the two problems which he proposed in the above section to consider at a future period.] . POissoN. 97 118. In the remaining three sections of the memoir Poisson discusses an example. In an addition to the work entitled MetJiodus inveniendi Uneas Euler determines the figure of the elastic lamina, properly so called, by means of a principle commu- {ds nicated to him by Daniel Bernouilli, namely, that the integral —^ J p taken throughout the length of the curve should be less than for any other cm've of the same length ; ds being the differential element of the sought curve and p its radius of cur\^ature. In order to give an example of the employment of the preceding formula, we will extend this principle by induction to the figure of equi- librium of an elastic lamina which is curved in every direction and the points of which are not acted on by any given force. Thus denoting by p and ^ the two principal radii of curvature at any point of this sm-face, or more generally the radii of curvature of two normal sections at right angles, and by da the differential element of the sm-face, we shall suppose that among all surfaces of the same area the elastic surface is that which gives a minimum value to the integral \\\~ -^t) ^°'- [This is what Poisson says, but he really takes the integral \\i- + -A dxdy , the two however coincide to the order of approximation which he finally preserves.] By the theory of the curvature of surfaces we know that the sum -+-C. has the same value for every pair of normal sections at right angles passing through the same point. With the notation already adopted, we have z. 1 1 _ (1 + z^) z" - 2z'z^ z; + (1 + g'" P ?~ (l+2'^ + O^ or, which is the same thing, 1 1 p ^ where u = , = , v = ■ ■ , ... ' == . ^I\■\■z^ + zf ^l+z" + zf 98 poissoN. We have also c?o- = Vl + z^ + zf dx dy. Let c denote an undetermined constant, and put then the question amounts to making the integral JJVdxdy an absolute minimum. (See Art. 104.) The quantity N of the nineteenth section (Art. 102) will be zero, and P, Q, B, S, T, will have for values ^ , , , fdu dv,\ ^ ^ / I \ fdu dvA T» ^ / » \ fdu dv\ ry ^ / / \ f du dv, «=2(«' + .,)(^ + - ™ ^ , , V ( du dv, It will be sufficient to substitute these values and their first and second differential coefficients with respect to x and y in the equation JT=0 of the twentj-first section (Art. 104), in order to obtain the indefinite equation to the elastic suiface ; this equation will be a partial difi'erential equation of the fourth order. We must also substitute these same quantities in the equations of the twenty- sixth section, in order to obtain the equations relative to the peri- meter of the elastic surface in all the cases which can occur. We will confine ourselves to writing these equations for the case where the elastic surface differs but little from a plane figure parallel to the plane of x and y ; and we shall neglect consequently the terms in Fof the fourth degree with respect to partial differ- ential coefficients of z. Thus the values of P, Q, ... and therefore the equations in question will be exact as far as quantities of the third order. I POissoN. 99 Thus we have, simply V= {z"+ zj + 2c + c (s'2+ «/), from which we obtain iV^=0, P=2c/, Q = ^cz^, R=T=2{z"^z^), S=0; thus the equation iZ = will Ibecome z""+ 2zJ'+ z^^^^ - c [z" + z^) = 0. If we denote by ^ a new variable, we may replace this equation by the following system of two equations of the second order: z"+z,=^, r+?. = cr (a). In consequence of these values of i?, S, T, the quantity Z of the twenty-fourth section (Art. 107) will be equal to 2^. In order to fix our ideas, I will suppose that the limits of the elastic surface in equilibrium are curves fixed and given, but that the tangent plane to this surface is not restricted by any condition throughout the perimeters of these cm-ves ; hence it will follow from the second case of the twenty-sixth section (Art. Ill), that we must combine with the two equations of each limiting curve the equation Z=Q or ^=0, in order to form the two systems of simultaneous equations, which with the given area of the elastic lamina will serve to determine the constant c and the arbitrary functions con- tained in the integrals of equations (a). The area of the lamina cannot differ much from that of its projection on the plane of X and y; denote the area of the projection by A,, and that of the lamina by X(l + ^) so that^ is a very small positive fraction; we shall have \ (1 +^) = U^Ji + z'^ + zf dx dy, or to that order of approximation which we ha.ve adopted ^9 = \\\{^"+^!)dxdy (5). 119. We may give another form to the equations {a) and (b) by changing the rectangular co-ordinates into polar co-ordinates. Let r be the radius vector of the projection of any point of the 7—2 100 POISSON. surface upon the plane of x and 3/, and 6 the angle which this radius makes with the axis of x, so that x = r C0&6, y = r &va.d. The ordinate z will become a function of r and 6, and we shall have -r =z cos ^ + s, sin ^, -ja = z^r cos ^ — s'r sin ^ ; hence , dz ^ dz Bva.6 dz . n dz cos 6 '=d?''''^ + dd-^' and as the element dx dy will be replaced by rdr dO, the equation (&) will become ^=i//ey-Mi)]^^'<^^ •.••... (^1. If we put s' in the place of z in the value of z , we shall have ,, dz ^ dz sin 6 by differentiating the value of s' in succession with respect to r and 9^ we obtain dz _ d^z d'^z sin dz sin B d^~'d?'''^^ dTdd^r^del^' dz _ d'^z -. d^z sin 6 dz . ^ dz cos 'dd~"dFTd''''^^~w '~r~'dr^^''^~dd~r' hence ,,_d'^z 2^ c?^3 sin ^ cos ^ ^^'^a; sin""^ ^ dz sin^ 9 dz sin ^ cos 9 dr r d9 r^ POISSON. 101 We shall find in the same way that «^'^ • 2/j o ^'^ sin g cos ^ , d^z cos'^ ^.= ^ «i^ ^+ 2 ^^ - + ^Q, ^,. dz cos^^ ^ sin 6 cos ^ c?r r dd r^ ' The same transformations will apply to the differential coeffi- cients ^" and ^,, ; thus the equations (a) will be changed into the following : d^z 1 d^z '^ dz _ y, -\ 'd?'^?~d^^T''dr~^ I ). (d). From the hypothesis of the preceding article and the supposition that the exterior and interior curves which hound the required sur- face are determined by the same equations as those in the twenty- eighth section (Art. 117), it follows that the value of z which we shall obtain by the integration of equations [d) must satisfy simultaneously the three equations r = F{e), z = ^{e), ^=0 (e) relative to the exterior limit of the surface, and must satisfy simul- taneously the three equations r^fid), z = cf>{e), ^=0 (/) relative to the interior limit. In the most usual case this second limit will not exist ; according to what has been explained above we shall then replace the equations (_/) by the condition that the value of z, which corresponds to r = 0, shall not become infinite ; and the same must hold with respect to ^, since we have supposed in the preceding article that the partial difierential coefficients of z, and therefore ^, are very small quantities through the whole extent of surface which we are considering. In order that there may not remain any doubt on this last case I will complete the investigation on the simplest hypothesis, namely, supposing that the elastic lamina is circular and that its figure of equilibrium is that of a surface of revolution. 102 POISSON. 120. If we take the axis of the surface for that of z, the quantities ^ and z will be independent of 6, and the equations {d) will reduce to ^ i^_c, ^ , i^_ J- / >! Let a denote the given radius of the projection of the lamina on the plane of the co-ordinates r and 6 ; we shall have X = 7^a^ [It does not appear why a is said to be given ^ The double in- tegral contained in equation (c) will extend from ^ = and r = to ^ = 27r and r = a, and this equation will become dz , ' where /3 denotes the value of -7-; when r = a, or in other words the inclination of the tangent plane of the lamina to the plane of pro- jection at any point of the perimeter. When this inclination is given we can immediately deduce the value of I -{-g, which is the ratio of the area of the lamina to the area of its projection; and reciprocally. [It is difficult to comprehend this equation g = ^^^ ', the equation (c) is Poisson seems to put this = tt f -^ J rdr, which is not justifiable. However he only refers to this equation once again, see page 104. Moreover if he takes yS as given he has no right to the equation ^= at the limit; see the first case of the twenty-sixth section, Art. 110.] We may suppose that the plane of the co-ordinates r and 6 is that of the boundary of the lamina ; the equations (e) will then be r = a, z = 0, ^=0 (h). According to what I have found in another memoir {Journal de VEcole Polytechnigue 19^ cdhier^ page 475) the complete integral of the second equation (^) is ^ = a f V''^^'<=<'= "'do3 + h re''"^'"'°"-' log (r sin' w) Jco ; *' J n POISSON. 103 wtere a and h are two arbitrary constants, and e the base of the Napierian logarithms. It is indeed easy to verify that this value of f satisfies the second equation {g) ; for we deduce immediately Jf + if = oo/V'^"" COS'.,;. - ^ JV-v;™. ,,3 „^, + U re-''^'^"' cos' 0) log (r sin' «) da Jo _ Jl£ re-'s/^cos coswJft). By integration by parts we have — /'V'"^=°='- cosG)J« = - c [V'"^=''=°='' sin' &)£?« ; — f V**"^""^" cos G> log (r sin' w) cfw = ^ j^-r^/ccos,. g^g ^^^ - c j e-''^<=°^'' sin' w log (r sin' o>) ^g). Thus the preceding equation is reduced to %^j^\^=ac re~'"^'°"' dca + he [ V''^'~*=°=" log (r sin' w) Jw; dr^ r dr J ^ j o and this coincides with the second equation (c) by reason of the value of ^. I put 5 = and suppress the second term of the value of f ; otherwise ^ would become very large near the centre of the lamina and infinite at the centre itself. We have then simply ^= a fV'"^""'" do), or, which is the same thing, •' •'0 104 POISSON, By reason of the third equation (h) we shall have IT n Pg-aVrcoso, ^^ ^ PgaVJcos^o ^^ ^ Q .^ •'0 •' and if we replace ca' by another constant — 7^, we shall have for determining 7 the equation 1 cos (7 COS ft)) £?a) = (^*). •' The value of ^ will become r2 ry7' COS ft) , = a / COS -i (ift), Jq a. where - is put instead of a. I substitute this value in the first equation {g) ; then integrating we have dz__aoL f^^.^ryr COS CO do) aa^ [^ f^ ^^^ a cos ft) 77- j Q \ a / cos &> r dz dr come very large for very small values of r, and infinite for r = 0, we must have (7 = 0. For r = a we shall therefore have ^ aa [^ . . dco «« fS, / m p = — sm (7 cos ft)) 2 I 1 — cos (7 cos co) 7 in ^ cos ft) 7' J/ ^' ^^ 2 , '- cos ft) this equation will serve to determine the constant a, from the known value of /3 or V2^. Integrate again, and denote the arbitrary con- stant by /; thus we shall have for the equation to the required surface -f-fl> 7r cos ct)\ dr cos ' a j r d(o cos (o aa f 2 dw —2- 11 — cos (7 cos w)\ — 5— 7 Jo cos^o) We can replace by convergent series the definite integrals which occur in these different formulce. In this manner the equation {i) will become >-^'+(oy^- (1:1:3)'+ •■•=*' <*)' where 27 has been put for 7. The values of 7^ which can be de- duced from this equation are known to be infinite in number, and all real and positive ; the least of them is, very nearly, 7^=1-46796491. This number, which occurs in several problems, has been cal- culated by M. Largeteau, secretary of the Bureau des longitudes. [Poisson gives no reference with respect to the roots of the equation just considered; the statements are proved in the memoir by Fourier entitled Theorie du Mouvement de la Chaleur. Mem. de VAcad. Tome IV. 1819, 1820, page 432.] We shall have at the same time • Traf 7V 7V 7V ^ 2 [ d' "^(1.2)'a* (1.2.3)'a«"^' dz _ irar ( rfr^ fyV* 7°/ ) ^ ~ T" I 2? "^ 3 (1.2)^ a* ~ 4 (1 . 2 . 3)^ a« "^ • ••)' _ Wf T^ _7V 7V' 1 J^ 8 1 4a' "^9 (1.2)^ a* 16 (1 . 2.3)'^a«'^ *••!• k 106 POISSON. The equations on which the values of a and / depend will be Trace f 7' _^ 7' , ^ ~ 4 1 2 ^ 3 (1 . 2)^ 4 (1 . 2 . 3)' "^ ■ 2 /■ 2 4 6 ^ "-/+ 8 t 4^9(1.2)^ 16(1.2.3)'^ J ' [thatis,/+^ = 0b7(Z.)]; this shews that cceteris paribus the sagitta / will be proportional to the radius of the lamina a ; if we take the smallest value of , "^ J IT 4 r"^ F' {ry) = / sin (27 cos©) cos 6) 4, g ( ^^® ^^*^ formed on the same supposition, namely, that dy = 0. It is evident that ^ , dBz , dBx dSy „^ , ,5. , d{Bz - z'Bx - z^By) We shall obtain in the same manner s 's , s , d{B z-zBx-zBy Bz^ = z^Bx + z^^By + -^ 8—2 116 OSTROGRADSKT. For the differential coefficients of tlie second order we have gg" = ^ (^' + ^^') _ ^" d {x + Bx) " ' -^ , _ d{z' + Bz') , _ d{z^ + Bz) , ^^•-d[y + Bij) ^'~d{x+Bx) ^" 5, d{z^ + Bz) Then we shall obtain the variations Bz", Bz', Bz^^ by changing z into z or z^ in the values of Bz' and Bz^. Thus we shall have 8." = z"'Bx + zl'By + d^^''-'"^^-<^y) , Bz: = <'^^ + z\,By + d{^^'-^Jx-z:By) . ay Therefore cZ^ (S^! — 2;'Sa; — z^y) Bz^ = z^'Bx + z\^By + dxdy Bz,=z\Bx + z,By + ^^^^'-''^f-'^y\ " " '" ^ dy^ And similarly we can find the variations of the differential coefficients of the higher orders. [These results agree with Poisson's; see Art. 102.] 125. The preceding method shews sufficiently how by direct application of the characteristic B to the partial differential coeffi- cients z, z^, z", ... we can find the 'variations of these differential coefficients. But it is better to seek the variations Bz', Bz^ , Bz", . . . by the use of total differentials. OSTROGRADSKY. 117;; In order to consider the subject with due generality, let us designate by w a function of as many quantities x, y, z, ... as we please, and suppose that the variable u and the independent quantities x, y, z, ... receive simultaneously the increments hu, hx, Sy, 8z, ... which we shall consider as arbitrary functions of all the independent variables. In order to find the variations ^du ^dit ^du dx' dy^ dz ^ due to the increments Sic, Bx, By, 8z, ... let us take the funda- mental equation Bdu = dBu ; put for dBu its value dBu , dBu , dBu , —dx+-^dy + ^dz + ..., and for du its value du J du J du y develop Bdu, that is, -> fdu -, du ^ du I , \ in the following manner ; - -, f^du du dBx du dBt/ du dBz \ , Bdu = [o^-+-j' -7— +:7--T^+ T- -J-+ ... jdx \ dx dx dx dy dx dz dx J /^ du du dBx du dBy du dBz \ , \ dy dx dy dy dy dz dy "' } ^ jL.(?, — j^~ <^8a; du dBy du dBz '\ , \ dz dx dz dy dz dz dz '"J + ...; 118 OSTKOGRADSKY. Now equate the coefficients of tlie arlbitrary quantities dx, dy, dz, ... and we have p.du _ dBu du dBx du dhy du dBz dx dx dx dx dy dx dz dx *"' ^du _d8u du dBx du dBy du dSz dy dy dx dy dy dy dz dy '"' ^du _ dSu du dBx du dBy du dBz dz dz dx dz dy dz dz dz '"' It is easy to give to these expressions the following form ^du _ d\ 5j d^u ^ d% -, ,dx dx^ dxdy " dxdz + J f^ du ^ du ^ du ^ \ a \hu — ^hx — T-oy — rr-hz — . . . V dx dy dz J dx ^du_ d\ ^ d\ 5j d\ ^ dy dxdy dy^ ^ dy dz '^ '" J f^ du -> du ^ du ^ \ d\ou — ^hx — T-ou — J- bz — ... I V dx dy '' dz J _^ % dy -, du d u ^ d u ^ d u ^ , o -7- = -= — 7- ox + -i — 7- by + -7-2 bz + . . . dz dxdz dydz ^ dz , / « du ^ du ^ du ^ \ d\bu — -=r^^ — j-by — -T-bz — ... V dx dy dz / dz For abbreviation put r> du rv CLU ^ 0!?^ r> . , T\ bu = -y- bx + -^by+ -y- bz + ... + l)u : dx dy "^ dz , tv du d\i ^ d\ ^ d\i rv dDu thus o -7- = -7-3- bx + 7 — =- by + -7 — ^ bz ^ ... ■\ = — , dx dx dxdy ^ dxdz dx OSTROaEADSKY. 119 ^du d\ ^ d^u ^ d^u ^ dDu o-T-= 7 — i-ox+ -7-2- OV + -J— J- oz + ... +-7— dy dxdy dy^ ^ dydz dy -> du d^u ^ d^u 5. , d\ 5, , , dDu o^r = "7 — 7- OX + -J — 7-0?/+ -7-2 OS + ... H — -7 — , dz dxdz dydz '^ as as We may remark that the terms wliich do not involve Du in the preceding formulae are the ordinary differentials of the quantities du du du -IIP,- ^ 1 u, -J- , -J- , -J- i •" considered as functions 01 x, y, z, ... and sup- posing that the differentials of x, y, z, ... are Sx, By, Sz, ... If then we denote by the symbol A the differential of a function of a;, y, z, ... due to the increments Bx, By, Bz, ... we shall have Bu = Am + Du, ^du _ . du dDu dx dx dx ' ^du _ . du dDu dy ~ dy dy ' ^ du _ . du dDu dz dz dz ' It is not difficult to find the variations of the higher differential dhi dSt coefficients -r^ , , , j • • • ; it may be easily seen that we shall have generally, dhi . d'u d'Du dx' dy"' dz\.. dx' dy"" dz^...^ d^ df' dz""...' [The method of this Article appears less clear than that in Art. 124 ; there is a want of definition of what is meant by such a symbol as S -7- . In the first method definition is given and consequences deduced from it; the formulae given in the present Article may be obtained by the first method. An additional advantage in the first method is, that we can see more easily to what order of approximation the results are true.] k 120 OSTROGRADSKY. 126. What has now been given will suffice for finding the variation of a function U which involves u, x, y, z,... and the diiFerential coefficients of u with respect to the variables. We have only to take the differential of Z7 supposing that all the quantities X, y, z, ... w, -7-,... receive their variations denoted by the symbol B. But since the variations of each of the quantities u, -^ , ... is composed of two indefinitely small quantities, we may by the principles of the Differential Calculus augment x, y, z, ... cL'Uj by Sfl?, hy, Bz, ... and give to u, -j- , ... at first only the former parts of their variations, namely Am, A -y- , . . . Thus we shall ob- tain an increment for U which will form the first part of the variation S U. Then without changing x, y, z, ... we can augment u and its differential coefficients by the second part of their variations Du, — ^ — , . . . ; the increment which the function Z7 will inconsequence receive will form the second part of the variation of U. The first part of the variation h U will evidently be ax ay "^ dz where -^ means the com'plete differential coefficient of TJ with re- spect to a;, and -7- the complete differential coefficient with respect to y, and so on. Let us denote hj DU the second part of the variation h U; this part is due to the increment Du of the quantity u, this increment being ascribed to u wherever it occurs in U. We shall then have SU=^Bx + '^J^B,/ + ^Bz + ...+DU. dx dy '^ dz We abstain from writing the development of the differential DU. OSTEOGKADSKY. 121 127. Let us now proceed to find the variation of tlie definite integral V= I Udxdydz ... taken for all the values oi x, y, z ... which satisfy the inequality L ^ „ (dZ /b 1-7-.. We shall obtain in the same manner jry \dZ J and so on. The denominator of the last differential will be unity; for if, for example, Z were the last variable, we should have had dZ=-^r-dz. dz Now form the product dX. dY .dZ ...', we have dX.dY.dZ ... — /Sf-^— .-y— .-^-... ) dxdydz ...I \ax dy dz J "^ therefore h{dxdydz ...) = \S\-r-' -j- >-j- ...\ — Vfdxdy dz .., The principles of Differential Analysis require that in calcu- lating the coefficient ^fdX d^ dZ \_ \dx ' dy ' dz '" ) we should take account only of infinitely small quantities of the first order, because j _. j — '— is an indefinitely small quantity of the first order. But except the term -7- . -7- . -y- . . . all the terms ^ dx dy dz 124" OSTROGRADSKY. in g[ — ,_^, ,,.) are of the second order at least; thus the \dx ay dz J following is true as far as quantities of the first order: fdX dY dZ \ dX dY dZ \dx ' dy ' dz'" j dx' dy ' dz Hence h {dx dydz...) = { -^ -y- -y- ... — \\dxdy dz ... Eestore for X, F, Z, ... their values a? + Sa?, y ^-^y, z-\-hz,...', we shall then have Therefore retaining only small quantities of the first order ^,777 . fdhx dBy dBz \ [dx dydz...) = \^ -^ [- -—- + -7— ■\- ... \dxdydz ... [This result may be simply found as follows; suppose for example three variables, and take the equations 7 -^j- CuJL 7 ciJL 7 dJL . dX = -j-dx + -7- dy +-7— dz, ^ dY , dY J dY. dZ 7 dZ 7 dZ 7 ^^d^'^'^'^'dv^^'^d^^^' tVtAj UbU U/iQ The second and third equations shew that dy and dz are of dY dZ the second order compared with dx ; for -7- and -y-^are indefinitely small while -3- and -y- are finite. Hence if we reject terms of dy dz •' the third order dX = -y- dx. dx Similar equations hold for e^Fand (^Z; therefore dXdYdZ— —Y- . -^r-' ^— d^ dy dz dx dy dz "^ OSTEOaRADSKY. 125 where terms of the second order are rejected. Thus With respect to some of the points suggested by this article the student is referred to Chapter xi. of the Treatise on the Integral Calculus.] 129. Before proceeding farther we will determine the limits of the variables x, y, z ... in the integral I Udx dy dz when extended to all the values of x, y, z ... which satisfy the inequality i < so that at the limits of the integral we have L — Q. We propose to integrate first with respect to x, then with respect to y, then Avith respect to z, and so on. Assume that the equation X = when solved with respect to x gives only two values for this variable x^ and x^. These values are the limits of the variable x, and supposing that the function L continues negative for values of x comprised between x^ and x^, we must integrate the expression I Udx dy dz from x = x^ to a? = a^i , supposing x^ less than x^ . As to the quan- tities y, z, ... we must ascribe to tliem all values which allow x^ and rKj to be real, and we must exclude all values which make x^ and x^ imaginary ; but in passing from real to imaginary values the roots x^ and cc^ become equal, as we know from the theory of equations ; therefore at the limits of y, z, ... we shall have simul- taneously dL If we eliminate x between tliese two equations we shall obtain an equation in y, z, ... ; this equation we will suppose gives two values of y, say y^ and y^ , which will be the limits between which we must integrate / Udx dy dz ... with respect to y ; we take the integral from the less of the two values ?/q and y^ to the greater. 126 OSTROGRADSKY. We shall arrive at the same result in the following manner ; after having integrated with respect to x we ought to integrate with respect to y obviously from the least to the greatest value of this variable, supposing x and y connected by the equation i = 0, and considering z^ ... as constant; differentiating on this hypo- thesis we have dL dL dy = 1 — • dx dy dx'' in order that y may be a maximum we must have -^ = 0, and this dL gives to determine the limit of y the equation -j- = ; this coin- cides with the result already found. dx dx To obtain the limits with respect to z we must treat the equa- tion which results by eliminating x between L = and -^- = precisely as we have already treated the equation L = 0. But we may suppose that this result of the elimination of the variable x between L = and -j- =0 is the equation L = 0, in which we put dL for X its value found from i— = 0. In order then to find the dx limits of z we must differentiate the equation L = with respect to y, considering a? as a function of y ; this will give dL dL dx _ dy dx dy ' and therefore -y- = since -j- = 0. By eliminating y between dL L = and -7- = we shall obtain an equation which will furnish the limits for z. By proceeding in this way we shall find the limits for all the variables which occur in the integral i Udx dy dz ... Thus we have the following conclusion ; the limits of x are given immediately by the solution of the equation L = with OSTEGROADSKY. 127 respect to x\ the limits of y are detei-mined by solving with respect to y the equation which results from the elimination of X between L=Q and -7— = ; the limits of z are determined by solving with respect to s the equation which results from the elimination of x and y between and so on. T —(\ ^^ A ^^ ^ dx ^ dy ' We have supposed that the equations relative to the limits of the integTal / Udx dydz ... give only two values for each of the quantities x, y, z, ... but it would be easy, from what has been given, to treat the case where the equations have more than two roots. The number of limiting values for each variable x, y, z, ... including if ne- cessary infinite values, must be an eveji number. 130. We now return to the variation + \DU dxdydz . . . ; for shortness put Uhx = F, UBy = Q, USz = R, ...; we shall have then BV=jff+^ + ^+...]dxdydz... +JDUdxdydz..., Consider first the part f(dP dQ d_R \ of the preceding variation ; suppose that x^ is the gi'eater of the two values x^ and x^^ which are obtained by solving the equation L = with respect to x. We have j-^dxdydz... =j{P^-P^)dydz 128 OSTROGEADSKY. By Pj we denote the value of P when x^ is put in it for x, and by P^ the value of P when x^ is put in it for x. Since the function L has a positive value before it vanishes when a? = a?o, and a negative value before it vanishes when dL x = x^, it follows that the differential coefficient -^ is negative for x = Xq, and is positive for x=x^; therefore if we take the radical \/ {^1-2] positively we shall have p dx \/[dx'J when x = Xq, p dx \/{dxy when x = x^. Substitute these values in the equation j^Jxd,jd.... - \{P.-P:)dydz. J and we shall have \-^dxdydz... = I -—fijf^dydz the integral on the right-hand side includes only those values of x,y^ z, ... which satisfy the equation L = 0. In the same way we shall obtain dL I -J- dxdy dz ... = / — i ^j^ - dx dz \-^dxdy dz ... = I / /^t-^\ dxdy ... OSTROGEADSKT. 129 Thus, doc dydz... + I -^j^dxdz... r B^ dxdy ... + .... The integrals on the right-hand side must be taken for those values of cc, y, z, ... which satisfy the equation L — 0. Consider two of these integrals, for example, P— C — jj^dydz... and —^dxdz...', from the preceding article we may easily see that their limits with respect to z, ... are the same ; further we have dL -, dL -, ^clx+-^dy = for all the elements of these integrals in which the variables z, ... remain the same ; so that the differentials -y- dx and -j- dy are equal, neglecting the sign. Thus if we take the increments dx and dy positively and also the radicals, we have dy dx and therefore multiplying \>j dz ...^ dy dz ... _d'' dz ... T^\ " /(^\ ' [dx'J V\df) - ■ 130 OSTROGEADSKY. It is easy to deduce tliat in general we shall have dydz... _ dxdz ... _ dxdy ... _ ^J\d^) vw) \/['d?) Hence, if for shortness we put ds = ^{dy'dz' ...+dx'' dz' ... + dx'df ... + ...), dydz ... dxdz ... _ dxdy ... __ dn\ ^ /idL^\ /Y^ dxV y\df) \/[dz') ds dU dU dU dx^^ df"^ dz""^'" By means of these ec[tialities, equation (A) will become T^ dL ^ dL T, dL , V 7 dx dy dz ' ■J (dJI dU^ dU Ua?' "^ dy^ ^ dz" "^ We may, in order to facilitate the integration of the differ- ential -r, dL ^ dL „ dL \ 7 dx ay dz J /(dU dU dU \ ' \l \dx^'^ df ■*■ d£' "^ '") instead of the variables x, y, z, ... connected by the equation Zr = introduce other variables a,h,c, ... which are independent. We must transform by the usual method all the elements dydz..., dxdz..., dxdy..., into elements proportional to the product da db ... ; we shall have such results as dy dz ... = A da db ..., dxdz ... — B dadb ..., dxdy ... = Cdadb ..., where A, B, 0, ... , are finite functions of a, b, ...; thus ds^dadb ... V(-4' + 5'+ C^'+...). OSTEOGRADSKY. 131 If for example we wish to integrate with respect to the variables y, z, ... we must observe with respect to the elements dxdz ..., dxdy ..., ... that we must take the differential of the variable x in the first considering y alone as variable, in the second considering z alone as variable, and so on ; hence dx dz ... = -J- dy dz ..., dxdy ... =-^ dy dz ,.., ...] thus // dot? dix? \ ds = dydz...^{l-^-^, + -^,^....) /(dU dU dU /(dU V \dx\ so that (^)-/(f+f + f+--)^"^^^^- dy ,dL ^dL T^dL ( -r^dh r^dL T~,dL ^ 7 7 [p^Q-,^^^dyd^ fdU\ \dx') In the formula (B) restore for P, Q, E, ... their values USx, USy, U8z, ... ; we shall then have 'U(-r-Sx+-r-8v+-T-Bz + ....] ds \dx dy '^ dz / that is, /(dll_ dU dU \ V W '^ dy-" ^ dz"" ^"") J I CtiX/ CtTJ CC^ J USLds dU dU dU dx^ dy"^ dz' 9—2 132 OSTROGRADSKY. and therefore «^= [ J,dL' '^u'^dU . +\DUd.dyd..... 131. "We now proceed to indicate the reductions to be made in the term jD Udx dy dz ... of the variation h V; these reductions consist in making as many as possible of the partial differential coefficients of the quantity Du disappear under the integral sign. By means of the formula (B) of the preceding article it will be easy to replace the integral JD Udx dydz... by the sum of the two integrals jWDudxdy dz ... and J@ds, the first of which like JDUdxdydz ... relates to all the values of x, y, z, ... which satisfy the inequality L <0, and the second of which comprises only those values of the variables which satisfy the equation L = 0. The function W does not include the variation Du ; on the other hand, the function © does include it as well as its partial differential coefficients with respect to x, y, z, ... ; the differential ds is the same as in the preceding section ; that is, ds = i^idy^d^ . . . + dd^dz^ . . . + dx^dy^ ... + ...). Thus we shall have I I) Udx dydz ... = j WDu dxdydz... + 1 ®ds ; and therefore hV=\wDudxdydz...+ ( ^dlf" dU N+/®^^- ^ \ \dx''^ dy'''^ dz''^'") The integrals I WDu dx dydz ... r uhLds and dU dU dU dx' ^ dif^ dz"" "^ are not susceptible of any reduction, but the integral ^%ds may be still reduced. OSTKOGEADSKY. 133 To effect the reduction o£ JSds, we must first of all replace the variables x, y, z, ... which are connected by the equation L = by other quantities a, b, ... which are independent. The number of these quantities a, b, ... must be one less than the number of the original variables x, y, z, ... Now considering x, y, z, ... as functions of a, b, ... let us trans- form the element ds into an element proportional to the product dadb ...; we shall thus obtain ds = Kdadb ... where ^ is a finite function oi a,b, ... Let us also transform the Tpp ,. , ^ , , dDu dDu dDu d^Du d'^Du amerential coeincients — = — , — ^ — , —7—,.... .. „ , ^j — 7-,.... dx dy dz dx dxdy . , dDu dDu d^Du d^Du 1 n i ^ ,1 . mto —7 — , -^7— , .... , ., -, -^ — ^,....: we shall have lor this da db ^ da'' dadb end dDu _ dDu dx dDu dy dDu dz da dx da dy da dz da dDu dDu dx dDu dy dDu dz db dx db dy db dz db d?Du d'^Du dx^ ^ d'^Du dx dy dc^ dy? dc^ dxdy da da d^Du _ d^Du dx dx d^Du /dx dy dx dy\ dadb da? dadb dxdy\dadb db da) But since the preceding equations are not enough to obtain the 1 r 11 xi ^•^' dDu dDu dDu d^Du d^Du value of all the quantities -^-, -^ , -^, ...-^^, -y-^ , ... some of these differential coefficients will remain indeterminate; the others can be expressed in terms of these and of the quantities dDu dDu d'Du d^Du ^ ^ , ., . -^' -db^""~d^^ ^o^'-- ^^'*"^^ of considering some dDu dDu dDu ,d^Du of the differential coefficients dx ^ dy ^ dz ^ dx^ 134 OSTEOaRADSKt. , .... as indeterminate, it is convenient, for the sake of sym- metry, to introduce as many linear functions p, q, r, ... of — ^ — , dDu dDu d^Du d'^Du .,, , . , -^ — , — ^ — , .... , „ , -^ — ^, .... as will DC necessary m order dy ^ dz dx dxdy^ "^ „ , .... dDu dDu dDu d^Du to express all the quantities -^ — , -j — , —-r^ , ~d~^ ' d^Du . , p dDu dDu d^Du l^y ^^ *'^^' "^^' ^' ''' ' -^' ^T' -^^' -T—TLi J and it will be the quantities p, q, r, .... which will remain arbitrary. But the introduction of the quantities p, q, r, ... amounts to imagining among the variables a,h,... one variable more ca ; thus the number of quantities to, a, 5, . . . is equal to that of the variables X, y, z, .... Considering then x, y, z, ... as functions of a, a, h, ... we have the equations dDu _ dDu dx dDu dy dDu dz d(o dx don dy d(o dz dw "*' dDu dDu dx dDu dii dDu dz := I iL _j V ... da dx da dy da dz da "*' dDu dDu dx dDu dii dDu dz :=: I d. _j 1- ... dh dx dh dy dh dz db '"' d^Du d^Du dx^ ^ d'^Du dx dy dco"^ do(? dcd^ dxdy doo dco '"' d^Du _ d^Du dx dx d^Du idx dy dx dy' dadw da? d(o da dxdy ydoy da da d(oj There will be as many of these equations as are necessary in T „ . dDu dDu dDu d^Du d'^Du order to express -^, -^ , ^, .... -^ , j^, ... m terms OSTEOGEADSKY. 135 p dDu dDu dDu d^Du d'^Du , ^ , . , ^ ^^ -^' -^' -^' •••• ^oT' ^^' - ' ^^* ^' *^" ^^^'^^^^^ '". really does not exist we must look upon tlie differential coefficients dx dy dz . . , . , ■^ , -T- , -J- , as quantities which we may assume at our 1 4. • rr xi • p dDu dDu dDu pleasure so as to simplify the expression of — , — , — ^ — , --^ — , ... Q/Ou O/U (X^ d'^Du d^Du rx^^ :i-a .- ^ «j . ^ dDu d^Du -M-^ l^f"" ^^" differential coefficients -^ , -^^,.... remain entirely indeterminate. Having expressed the differential coefficients dDu dDu dDu d'^Du d^Du dx ^ dy ^ dz ^ '" da? ' dxdy^ in terms of dDu dDu dDu d^Du d^Du d(o '' da ^ dh ^ " do)^ ' dcoda^ we must put these values in the integral fsKdadb.... l@ds=U Then by making use of the formula (B) and putting for short- ness ds' = ^/{dh\..+da\..+ ...) we can replace the integral JSKda db ...hj the sum l(pDu+Q'^+E^^+..?jdadb...+j^ds'. The first of these integrals is not susceptible of reduction ; the second may be reduced in the same way as J@ds. Thus we shall have BV=jWDudxdydz... + [ ^^' ^ -^ + j(PDu-i Q^+ R^ + ...^adb ...+ j^ds'. 136 OSTROaRADSKY. We may treat the integral J^ds' as we treated J®ds ; we may decompose it into two integrals, the first of which is completely reduced, and the second is susceptible of reduction; proceeding in this way we shall exhaust the reductions which can be effected in the integrals, and thus the variation B V will be in the proper shape for applications. 132. Since the integral J@ds of the' preceding article relates to the values oix,^,z,... which satisfy the equation L = 0, we may consider one of these quantities as a function of all the others, and the latter as independent. Suppose, for example, that we consider a; to be a function of y, ^, . . . ; we shall have (by Article 130) /fdT dU dU \ , \ \dx^'^ df'^ dz' ^ '"). -, ■~ — 1W\ — -^ V \dxV put for shortness Ifdn^ dJl dL" \ y — \ir {da?) we shall have l®ds={'^dy dz .., . We shall obtain the equation relative to the limits of 2^, 2!, ... by eliminating x between Z = and -y- = 0. (LdU The function ^ contains the partial differential coefficients dBu dDu dDu d^Du d^Du dx ' dy ^ dz '' '" dv? ' dxdy^ taken relatively to x, y, z, ... on the hypothesis that these variables are all independent; but kfter the differentiation we must put for x its value furnished by the equation X = 0. It is advisable to eliminate these partial differential coefficients as far OSTEOGEADSKY. ' 137 as possible ; to this end considering a? as a function of y, », ... we shall have dDu _ fdBvX fdDu\ dx dy \ dy J \dx J dy^ dDu _ fdDu\ /dDu\ dx dz \ dz J \ dx J dz^ d d fdDu\ \dx ) _ fd^Du\ /d^Du\ dx dy \dxdyj \ dx^ J dy ' f dDii\ \d^J _ f d'Du \ / d'Du \ dx dz \dxdzj \ dx^ ) dz ' d^Du _ / d^Du\ f d^Du \ dx (drDu\ d^ / dDu \ (Px dy' ~ [ dy' J'^ [dxdyj dy^\ dx"" ) dy^'^\dx) dy' d^Bu _ f d^Du \ f d'Du \ dx f d'Du \ dx dy dz \dydz) \dxdyj dz \dxdz) dy fd'^Du\ dx dx ' /dDu\ \ dc(f J dy dz \ dx J X dy dz \ dx J dy dz d'Du _ f d'BiA f d'Du \ dx / d'Du \ d^ f dBu \ d'x dz' ~ \d^) "^ [d^zj d^'^\dx' J dz'^\dxj dz' ' the brackets indicate partial differential coefficients of Du taken on the supposition that x,y, z, ... are independent. From the preceding equations we deduce the following, putting for abbreviation v for (— r-j and |-j-j for f , ^ j /dDu\ _ dDu dx \ dy J dy dy' /dDu\ _ dDu dx \ dz ) dz dz' 138 OSTEOaEADSKY. fd^Du\ _ dv /dv\ dx \dxdyj dy \dx) dy (d^Du\ dv (dv\ dx \dxdz) dz \dx) dz ' /d^Du\ _ d'^Du _ dvdx fdv\ dx^ d^x \ dy^ )~~df ~ dy'dy^\d^)d^~'^d^' d^Du\ _ d^Du dv dx dv dx /dv\ dx dx d'^x dy dz) dy dz dy dz dz dy \dx) dy dz dz dy d^Du\ d^Du ^ dv dx fdv\ da? d^x dz^ J dz^ dz dz \dx) dz^ dz^ ' dy'' dz'' '" dy^'' dydz^ dz^ '' '" their values found from the equation Z = ; thus dij (dDu\ dL dDu dL + V dx \dy j dx dy dy ' dL /dDu\ _ dL dDu dL dx\dz) dx dz dz ' dL /d^DvX _ dL dv dL fdv dx \dxdy) dx dy dy \dx. dL (d^Du\ _ dL dv dL fdv\ dx \dxdz) dx dz dz \dxJ ' /d^I>u\ _ dL (dl/^ d^Lu 2 ^ ^ ^ V dy^ J dx \dx? dy^ dx dy dy dx^ \ dy^ ) dx \dQi? dy^ dx dy dy dy"^ \dx) (dUd^_^dLdLd^ dUd^ \da? dy^ dx dy dxdy dy^ dx dL^ (d^Du\ _ dL idL^ d^Du dLdL dv dL dL dv_ dL dL fdv dx^ \dy dz) ~ dx \dx^ dydz dx dz dy> dx dy dz dy dz \dx \dU d^L dLdL d'^L dLdL d'^L dL dL d^L \dx^ dydz dx dz dxdy dx dy dxdz dy dz dx OSTEOaRADSKY. 139 dU fd''Du\ _ dL { so that the triple integral extends throughout the interior of a certain ellipsoid. Let the value oif{x,y, z) be changed by variation into where //- may be supposed indefinitely small, so that the varied triple integral extends throughout the interior of a new ellipsoid which is similar to the former and similarly situated and concentric with it. Then the part of BU which arises from the variation of y^ and y^ will be easily seen to be, not absolutely zero, but, an indefinitely small quantity, which may be called of the second order if that part which arises from the variation of z^ and z^ be called of the first order. Also that part of S f7 which arises from the variation 10 146 DELAUNAY. of x^ and x^ will Tbe an indefinitely small quantity which may he called of the third order. Thus it will be observed that by supposing the triple integral U to be extended over all the values of x, y, z which render /(a?, y, z) negative, the variation S Z7 is free from any terms arising from the variation of the limits of the integrals except those which arise from the variation of the limits of the first integration. This sim- plicity however is obtained by a corresponding loss of generality in the results. The most general supposition would be that the limits Sq and z^ are any arbitrary functions of x and y, that y^ and y^ are any arbitrary functions of x, and that x^ and x^ are any con- stants; so that no mention would be made of the function /(a;, y, z). One of the great merits of the memoir of Sarrus is that it treats the problem in this most general sense ; it will be remembered that Ostrogradsky had adopted the same limitation as Delaunay. (See Art. 129.) And in Poisson's researches on the variation of a double integral the same limitation occui'S, for the integrations are sup- posed to extend over an area bounded by a closed curve. (See Art. 105.) 139. We resume the consideration of the value of S C7" given in Art. 137. The last two terms in the value of SZJare united by Delaunay by means of a new notation, which he considers to pos- sess some advantage over that hitherto used ; in order to explain it he says he must enter into some details. Let //(^a; dyh he a double integral which extends over all the points in the plane of {x, y) comprised within the interior of the closed curve AmBn (see fig. 4). Let y^ and y^ represent the ordi- nates of the cm've which correspond to any abscissa x ; let x^ and x^ be the extreme abscissse Oa and Oh. Then the double integral may be expressed thus, j dxj dyh. Now suppose we have found the indefinite integral H of hdy, and let H^ and R^ represent what H becomes when we substitute y^ and y^ respectively for y. Then we have I dxj dyh=\ Edx-j HJx. DELAUNAY. 147 Now suppose that a point starts from A and moves round the cm've AmBn in the direction indicated by the arrows and finally retui'ns to A. Let dx denote the space traversed by this point in the direction of the axis of x in any indefinitely small time, and let y be the variable ordinate of the point. Then the integral jdxH taken during the time the moving point describes the por- tion AmB will form the first part of the integral jjdxdyh, namely, rxi I H^dx; J Xn and this same integral taken during the time the moving point describes the portion BnA will form the second part of the inte- gral JJdxdyh, namely, -j^'S^dx', J Wo for in this second part of the motion dx is constantly negative. We may then express the integral fjdx dy Ti completely by ^dx H ex- tended throughout the motion of the moving point, that is, from its departm-e from A until its return to the same point ; this will be indicated by the notation /, dxH, Besides the advantage of uniting in a single term the two terms which were required to represent the value of ^^dx dy Ji, the proposed notation has another advantage ; for we can express by a single term the integral j^dx dy h in the case in which the limiting cm-ve can be intersected in more than two points by a line parallel to the axis of y, as may be easily seen, 140. In the last two terms of the value found for h U (see Art. 137), we may consider the quantities K^z^ and — K^^z^ as forming the two parts of a definite integTal taken with respect to z between the limits z^ and z^. We may therefore, by Art. 139, put j 'dyK^ Sz, - r'dy K^Sz^ = i\^l/ ^^« J Ji/o J Vo .' {y) 10—2 148 DELAUNAY. thus 8U= dx\ dy \ dzX -t~ Sp I dx I dyKhz', + ' ^0 J {y) the z which enters into the K of the last line is a function of x and y determined from the equation f{x, y, z) — 0. 141. We must now transform the terms in the first part of BU by means of integration bj parts. This part of Delaunaj's memoir is treated by him with great generality ; his method will be easily understood from a simple example which we will take. Let^ stand for dx dy dz dK and If for , , dp then we have in the first part of S Z7 the term I dx\ dy i dz M-^ — ^ — j- ? J^o ho J!^o dxdydz and we will take this term and reduce it by integration by parts. By one integration by parts the term becomes dx dyM-j — =- - dx\ dy \ dz ^r- i~^r , J Wo J{y) dxdy ] x^ J y^ J zq dz dxdy where in the first term the notation is used which was explained in Art. 139. If we effect two more integrations by parts in the second of the above two expressions we shall easily see that we shall finally obtain in the indefinite part of the variation Sf/the term — I dx\ dy \ dz -^ — ^ — , Bu ; J xo J yo •> i^o dx dy dz we shall also have some limiting terms. The limiting terms it is not as yet easy to write explicitly, because the limits of the respec- tive integrations will not be the same as those we have hitherto DELAUNAY. 149 used, since the order of the integrations becomes changed. In fact to reduce the term [''^j [y^j f^i^ dM d^hit j ax\ ay \ da ^— -^ — r- J Xq J yo J zo dz dx dy as much as possible by integration by parts, we must begin by in- tegrating with respect to a? or y and not with respect to z, as we have hitherto supposed. But it will introduce confusion if we use different limits, and thus such a transformation is required of the terms at the limits as will allow the integrations to be all performed in the same order. This transformation, as Delaunay says, formed one of the principal difficulties of the problem, and he considers that he has accomplished it with all the simplicity desirable. He adds that Ostrogradsky had arrived at the same mode of transformation as a particular case of a more general method, this particular case being however the simplest that could be derived from it. See formula (C) of Art. 130. 142. Let for example fJNdydz be a term which has arisen from an integration by parts with respect to x and in which we have not yet taken account of the limits between which the integral is to extend, so that iV is a function of x, y, and z. In order to determine the limits we must deduce from the equation /(a?, 7/,z) =0 the values of x in terms of y and z ; suppose we thus obtain two values of X, which we may denote by x" ap.d x, and let N" and N' denote what iV" becomes when x" and x are respectively put for x; then the integral may be denoted by fJ{N" — N') dy dz, and it is to extend over all values of y- and z which make the values of x found from f{x, y,z) = real. We want now to transform this double integral so that it shall extend between the old limits ; and we shall now shew that it may be put in the form I ^dx I ^dyM, where M=N-S., dy and x^ and x^ are the same quantities as have been throughout denoted by these symbols. 150 DELAUNAY. For any assigned value oiy tlie equation /(a;, 3^, z) =0, may be regarded as the equation to a plane closed curve AmB of whicli the variable co-ordinates are x and z. (See figure 5.) Thus in the double integral we are considering, j^dy dz N, if we integrate with respect to z we must extend the integral to all values of z which allow us to deduce from the equation f{x, y, z) =0 real values of X, that is, the limits of z must be Oa and On. But by Art. 139, rOn c {Oa) {N"-N')dz=^ Ndz. J Oa J (Oa) • On r {Oa) {N"-N')dz=^ Oa J {Oa) Thus JJNdy dz takes the form • iOa) hi Ndz. (Oo) /, fiOa) The symbol I Ndz indicates an integral taken throughout the J {Oa) motion of a point which starts from A and returns to A again after moving round the cm've in the order of the outside arrows. But if we suppose a second point to start from B and to move round the cm-ve in the order of the inside arrows and return to B the symbol (06) Ndx would indicate an integral taken throughout the second (05) motion. But dz and dx being the increments of z and x which cor- respond to the instants when the moving point is traversing in opposite directions the same element of the curve, we have obvi- ously dz —— y- dx, dz . where -7- is the differential coefficient of z with respect to x de- duced from f{Xj y, z) = 0. Thus r(Oo) r(06) J„ dzN=- dx^N. J (Oa) J (Oh) "^ Therefore the double integral we are considering becomes J J { (Ob) J^ dx-j-N. iob) dx DELAUNAY. 151 It must be observed tbat in N in the last expression z is to be considered a function of x and not x of s. This definite integral extends over all the values of x and 3/ which allow of real values of z being found from f{x, y, s)=0, as is easj to see; and as the order of integration may be changed at pleasm*e, we may take that which has already been adopted in Art. 135. Thus we have finally ffNdi/ dz, that is, ff{N" - N') dy dz ^-{''dx\'%^N=\''dx\'''dyM, J^o hy) dx Jxo hy) where M== N-r-s . aj dz 143. We shall not reproduce the extremely general formulss which Delaunay now gives with respect to multiple integrals, which extend over pages 59 — 73 of his memoir. His method will be suf- ficiently illustrated if we give in detail the investigations of the variations of a double integral and of a triple integral, in which we shall suppose that no differential coefficient of a higher order than the second occurs in the proposed expression. Let us then consider first the variation of a double integral. Let 'u=\'''dx\^'dyV. J Xn J Vn _ __ , . . -, du du d\ d'^u , Let V be a function of a^, 2/, ^^, ^. ^^ ^^ ^^ ^^^ 72 ^ ; and let the variation of CT'be required arising from a varia- dxdy tion in u and a variation in the limits of the integrations. The partial differential coefficient of V with respect to -y- will be denoted by F,, that with respect to -j-^J ^y? ^^^ "^ith respect to -^ by F^^ , and so on. 152 DELAUNAY. Then, as in Art. 140, J xo J yo J (x) _dV ^ p. dhu j^ dSu p. d^Su p. d^Su j^ d^Bu ~1^ '^ ""^"^ "'dy^ =^^~^"^ ^^'l^y^ ''"'df' thus there are six terms in SF; and we shall consider how these six terms will appear in 8 TJ. The first term is not susceptible of reduction. The second term is / \dxdy V„-j—; loj Art. 142 this gives — I dx-^ V^Bu — I jdxdy-j^Bu. The third term is lldxdy Vy-j- ; /•(«) rr ^Y this gives I dxVyBu— lldxdy -~Bu. J (K) J J ay The fourth term is tldxdv F, '^^ dx"" ' by one integration by parts this gives is lldxdy dBu and by a second integration by parts we obtain p) Jy „ dBu r'^' dydV^^^ , [[, , {''..-(iy-"-^-i}?-/;>^- 144. We shall now give the variation of a triple integral. Let Z7= 1 1 \dx dy dz V, that is I dx i dy I dz V, JJJ J Xq J i/o J Zq V is supposed to contain x, y, z, u, and the partial difierential coefficients of u with respect to x, y, z, up to the second order inclusive. Here BU=jjjdxdydzBV+j'''dxj^^dyVBz; y d^Bu „ d^Bu p. d^Bu y. d^Bu __ d^Bu p. d^Bu "r ' as _7_-2 r ' »/u 72 r ' 22 7 2 r ' x?y 7 7 r ' arz 7 7 r ' »> "* ^a;^ ' "" dy' ' ^^ c^s'' ' ''^ dxdy ' '^^ (^a;dy Tx^J^- jlj'^ 'iy <>■' dx Ac ' by a second integration by parts we obtain -jjdxdy^£ r^'^ + jjdxdy^^Su + jjjdcdyd,^' Bu. The sixth term is jjjdx dydzvj^; by one integration by parts this gives {{j J ^^ IT ^^^ [ffj 1 J dVyydBu -jjdxdy^ V^^-jjjdxdydz -^ -^y, DELAUNAY. 157 by a second integration by parts we obtain - jjdxd!, I F„ -J + JJ) dx, where m is some function of a? at present undetermined. Delaunay considers that there will be different cases in this problem according as the differential coeffi- cients which occur in <^ are, or are not, of a higher order than those which occur in K. If, for example, the highest differential coeffi- cient which occurs in <^ is one order higher than the highest which occurs in K, Delaunay arrives at the result that at each limit of DELAUNAY. 161 the integration we must have m = ; if the highest differential coefficient which occurs in <^ is two orders higher than the highest which occurs in K, Delaunay arrives at the result that at each limit of the integration we must have m = and -z— = 0. 146. Without going into detail on the subject we will indicate two objections to Delaunay's conclusions. First. Suppose, for example, that K involves differential co- efficients up to the second order inclusive, and that ^ involves differential coefficients up to the fourth order inclusive. Let x^ and x^ denote the limits of the integi-ation, and suppose that x^ — x^ is divided into n equal parts ; and put x^ — x^ = nh. Then Delaunay says that the relation ^ = is meant to hold for the following values of x, when n is supposed large enough : x^ + Sh, Xg + Ah,..., Xq+ {n — 4) h, x^-^ {n — S) h ; that is to say, it is. not meant to hold for the values Xq, Xg+h, Xg + '2h, x^, x^ — k,Xj^—2h. . . For ^ involves differential coefficients of the fourth order, and such differential coefficients may be supposed to depend upon four con- secutive values of x; so that if for example we suppose ^ = to hold when x = x^—2h, a value x^+ h would be involved in ^, which lies beyond the limits of our integration. The reply is simple; the proposer of a problem may attach his own meaning to his con- ditions; he may say that ^ is to be zero for all values of x within the limits x^ and x^, or he may say that ^ is to be zero for all values of x within the limits x^ + 3A and x^ — S7i. Thus Delaunay's investigations do in effect attach one of two possible meanings to a certain condition, but probably not the meaning which would generally be attached to such a condition. Secondly. Let us now take Delaunay's own view of the mean- ing of the condition and examine if his conclusions hold. We have then the following problem: I ^Kdx is to be a maximum or a J Xq minimum while the condition ^ = is to hold for all values of a; comprised between ^„ and |^, where f^ and f, lie themselves be- tween Xg and x^. In Delaunay's problem the difference between 11 16^ DELAUNAY. Xg and ^0 is infinitesimal, and so is the difference between f^ and a?^j but we need not restrict ourselves with this limitation. We have then to make the variation of the following expression zero : I 'Kdx +1 \K+ mj>) dx + I ^Kdx. The variation as usual will consist of two parts, an integrated part and a part still remaining under the sign of integration. To make the latter part vanish we must take a solution which leads to discontinuity in the form of our functions; that is, a certain equation or certain equations will be obtained which must hold between the limits x^ and f^ and also between the limits ^^ and x^, and a certain other equation or certain other equations will be obtained which must hold between the limits ^^ and ^^. There will be no objection to this discontinuity in form provided we can also make the integrated part of the variation vanish ; this we must now consider. The integrated part which occurs at the lower limit of S I Kdx and the integrated part which occurs at the upper J Xq limit of S I Kdx may be made to vanish in the usual way by a proper disposal of the constants which occur in the integral of the differential equation obtained by making h\Kdx = Q. The inte- grated part which occurs at the lower limit of 81 \K+ni) dx will partly unite with that which occurs at the upper limit of S I Kdx • and the integrated part which occurs at the upper limit of B\ {K+ m(f)) dx will partly unite with that which occurs at the lower limit of B Kdx. Theoretically the complete set of terms at the limit 1^^ and the complete set of terms at the limit f^ can be made to vanish by a proper disposal of the constants which occur in the integral of the differential equjition obtained by making B!{K+m(f))dx = 0. DELAUNAY. 163 We must now examine some of these terms more particularly. We have abeady supposed y to denote one of the variables which occur in K\ put dy _ d?y _ d^y _ d*y _ d^~P' ^~^' 'M~'^' M~^' Then among the tenns of the integrated part we shall have and Sr and Zq^ will not occur elsewhere among the integrated terms. And as m is supjjosed a function of x only we have drn^ _ (?0 dm^ _ d^ ds ds ' dr dr ' Thus Br and Sq will disappear from the integrated part if we have, at the limits ^^ and |^j, d II I ll\ {y^—^y) d m ds {dy d^ dz d^y\ _ fds\^ dx fi dx \ds ds^ ds ds^ J ' anay."-.y'=(|y(|^-|g);tKus dz o^y _ '^ ^^^ f(^y ^^^ ^^ ^^y\ d m^ds /dy d^z dz d^y ds ds fJL da? \ds ds^ ds ds^J dx [jl dx\ds ds^ ds ds^ _dx d m /dsV /dy d^z _ dz d^y \ ds dx fi \dx) \ds ds^ ds ds^ J ' Divide by -j- and then integrate ; thus where G is an arbitrary constant. , . m /dsV /dy d^ _dz d^y\ _ m y'z" - z'y" ' fjb \dxj \ds ds^ ds ds^J ~ fx, ds dx fill I lis _ pm{yz -z y ) {l+y'^ + zy delaunay; 169 we may write equation (8) thus : a.z-l3y+ <^ = -^i^qryqi^Ty (9). Equations (7) and (9) are the integrals it was proposed to obtain ; the problem is thus reduced to depend upon the solution of equations (3), (7) and (9). The value of m from (7) may be sub- stituted in (9) and the result will, with equation (3), constitute two simultaneous differential equations of the second order for determin- ing the required curve. 150. "We have not yet verified Delaunay's statement that equations (4), (5) and (6) give a solution of the problem; we shall arrive at this result most easily by arranging the whole solution of the problem as far as it can be completed in a symmetrical manner. Take the arc s for the independent variable ; then we have /Jr»V , fdyV (dz^ ^ ^ fd^xV fdW (d^V- 1 ^ and subject to these conditions we have to make ds a maximum or a minimum. We may then in the usual way consider that we have to make I Vds a maximum or a minimum, where Here - and — denote fanctions of s at present undetermined. Hence in the usual way we obtain as the necessary equations for a maximum or a minimum, (10). d' ds' \' d'x ds' _d_ ds dx ds = o" d' ds' \ dSj ds' d ' ds ^Ts- = d' ds' \' d'z ds' _d_ ds dz ds = ^ 170 DELAUNAY. Therefore by integration i^' d'x ds' dx -"& = a ds d'y ds' ds h ' ds d\ ds"" dz ds ~ c _ (11). Multiply the first of equations (11) ^J -j- and the second doc by -J- and subtract ; thus dy d'^x dx d'^y\ d\' ds ds' ds ds .dy d^x dx d^y\ , _ dy ^dx ds \ds ds^ ds di) ds ds ' By integration we deduce the first of the following three equations, and the other two may be obtained in a similar manner, \ds di- ds ds^ J ^ '' ^ , (dx d'z dz d'x\ „, , (dz d'y dy d'z\ _ .(12). In these equations f, /', and /" are arbitrary constants. This method of solution is given by Mr Jellett; see his Calculus of Variations, page 195. 151. We have not yet considered the integrated part of the variation. We suppose that the extreme points of the cm-ve are fixed. Then with the notation of Art. 149 the integrated part of the variation will consist of dz' If we suppose that there is no restriction on the tangents at the limiting points, then since hy' and h^ are independent, we must have -T-?7 = and -7-77- = at both the limits in order that the in- dy dz DELAUNAY. 171 tegrated part of the variation may vanish. Hence from the known expressions for -j-rr and -7-77 we must either have m = at both limits, or else we must have simultaneously y + s'(^y' - y'^') = 0, and z" - y'{z'y" - y'z") = 0* at one limit or at both limits. But the latter equations are not admissible, for they lead by squaring and adding to y'- + ,'- + i^- + ,-) i^.y _ y',"Y + 2 {z'f - y'z'r = 0, and this would make that expression vanish which is ahyays equal to the constant - by hypothesis. 152. Let us now examine the form of the integrated part when we adopt the method of solution given in Art. 150. The integrated part consists of the following terms ; first, VBs, secondly, (^ -7 — ^^' -tt) (Bx — -r-Ssj together with two similar terms in y and z, thirdly, X' -j-^ (S -^ — -j-j- Ss) together with two similar terms in y and z. Since the extreme points are supposed fixed Bx, By, and Bz vanish; hence by using equations (11) we obtain for the inte- grated part /tt dx ^ dy dz \'\ ^ ^ ,(d^x ^ dx , d^y ^ dy , d'^z ^ dz\ where it is to be observed that F= 1. It will now be convenient to determine A.'; for this purpose dj^x d^ti d^z multiply equations (11) by -tt, -ty? ^ttj respectively, and add; thus 1 JV _ d^x , d'^y (Pz p^-''W^^'d?^''d?' 172 DELAUNAY. therefore -^ = a-^ + h^ + Cj- + Si constant ; p as as as and in order that the coefficient of Ss in the integrated part of the variation may vanish this constant must equal unity, so that V , dx ^ dy dz p as as as In order that the rest of the integrated part of the variation may vanish it may b,e shewn as in Art. 151 that A,' must vanish at both limits of the integration; this is proved by Mr Jellett on page 197 of his work. The value of X may also be found ; for this purpose multiply ci nr fill ft z equations (11) by y- , -^ , -y- , respectively, and add ; thus _ , idx d^x dy d^y dz d^z\ dx , dy dz \ds ds^ ds ds^ ds ds^J ds ds ds _ V dx , dy dz _ 2V p ds as as p 153. Now return to equations (12) of Art. 150 and remember the result just mentioned that X vanishes at both limits of the integration. Take the origin of co-ordinates at one of the fixed points ; then since we have simultaneously x = 0, y = 0, z=0, X= 0, it follows that /= 0, /' =0, /" = ; multiply equations (12) by ■J- , -^ , -T- , respectively, and add ; thus we obtain {ay -hx) ■£ + {cx-az) ■£ + {bz~cy) -£ = 0, ( dz dii\ , / dx dz\ I dy dx\ This equation may be integrated by assuming y = ux, z = vx', it leads to ay — hx = n {az — ex), DELAUNAT. 173 where n is a constant; this is the equation to a plane passing through the origin. Thus the required curve is a plane curve, and as a circle is the only plane curve of constant curvature, we obtain a circle as the required solution. 154. The preceding article is due to Mr Jellett ; it will be seen that it adds something to the result enunciated by Delaunay ; for Delaunay stated that a circle is a solution of the problem, while Mr Jellett shews that if there is no restriction on the tangents at the extreme points the required curve must be a plane curve and therefore a circle. Some further remarks however are necessary here. The pro- posed problem may be understood in two senses ; for we may be required to find a curve of maximum or minimum length while the curvature has soine constant value, or we may be required to find a curve of maximum or minimum length while the curvature has an assigned constant value. In the first case, when the curvature is merely required to be constant, we may take p as large as we please, and thus the solution will degenerate into the straight line joining the two given points. Let us next consider the second case, in which the curve must have an assigned constant curvature ; it might then be impossible to draw an arc of a circle so as to have a gi^^en curvature and to pass through two given points, and in fact this could not be done if the given points are at a greater distance than the diameter of a circle which has the assigned curvature. It becomes a question then, what the solution of the problem is in such a case where the distance of the given points is too great to allow of their being connected by an arc of a circle. We shall shew that the problem is solved by a set of arcs of the required curvature. Let A and jB be the two fixed points (see figure 6). Let A CD, DEF, FGB be three equal arcs of the assigned curvature, and let them be placed so as to have a common tangent at the points of junction. D and F; then we shall shew that the curve ^ (7Z)J5JF(r-S constitutes a solution of the problem under con- sideration. 174 DELAUNAT. For let us suppose the quantity s^ — s^ divided into three parts «i - <^ij o"i - ^o» <^o - *o ; then ["' Vds =!"'' Vds + r' Vds + f '' Vds, and the variation of the left-hand member will be zero if the variation of the right-hand member be so. Now consider h \ Vds. The part of this which remains under the sign of integration vanishes when equations (1 2) are satisfied. And by proceeding as in Art. 152 it appears that if V vanishes at both limits the integrated part at the lower limit will entirely vanish and the integrated part at the upper limit will reduce to — {ahx + i^y + chz) ; this term remains because the upper limit is now not a fixed point. The term just exhibited is destroyed by a similar term which occurs at the lower limit of h j Vds, if a, h, c retain the same values. J (To In this way we see that we shall have 8 f""" Vds + S f "' Vds + S f '' Vds = for the system of arcs in figure 6, provided that a, h, c retain the same values for all the arcs. It will be remembered that by Art. 152, -■='''(> + «S+*l-l) (^«)^ moreover the common tangent at D makes the same angle with AB as the tangent at B, and the common tangent at F makes the same angle with AB as the tangent at A ; thus if a, h, c retain the same values throughout the arcs, X' vanishes at F and B if it vanishes at A and I). Thus all that we have to do is to shew that equations (12) are true for all the arcs in figure 6 while a, h, c retain the same values throughout, and A.' has the value given in (13). That is, in DELAUNAY. • 175 effect we have to shew that equations (12) and (13) are true for a circle of radius p without imposing any restriction on the co-ordi- nates of its centre which it may be impossible to fulfil. Since the direction of the axes is in our power, let us suppose that A is the origin, AB the direction of the axis of x, and the plane of the arcs the plane of {x, y). Then 2 = 0; thus we must have /= 0, /' = 0, /" = 0, c = in (12), so that these equations reduce to ^, (dy d^x dx d^y\ . . . where V = ,'(l+4 + 5f) (15). Now let [x — hY+ {y — W —P^ ^6 the equation to the circle of which the first arc A CD is a portion ; and suppose that the axis of y is taken so that k is positive. We shall have from these we deduce dx . y — Tc dy _x — 7i ,^. ds p ^ ds p ds and supposing that s increases with x so that -^ is positive we must take the lower signs ; ,1 d^x 1 dy X — h ds^ p ds p^ ' d^y _ 1 dx _ y — h ds^ p ds ^ ■' Thus that (14) and (15) may be true, we require that o/, y — h,-,x — 7i\l , so that 1+.^^Z1^ = (17), P is the only relation between the constants. 176 ■ DELAUNAY. Now we have already supposed that V vanishes both at A and D; at tliese points -^ has the same value while — has values numerically equal but of opposite sign. We must therefore have & = 0, so that (17) reduces to aJc 1 + — = 0. P Now suppose h' and Jc the co-ordinates of the centre of the circle of which the second arc DEF is a portion. Proceeding as before we shall find that we must now use the upper signs in the equations which replace (16), and we shall finally arrive at l-^'=-0. P Thus 7c and 7c' are equal in magnitude and of opposite sign, and this is the only condition necessary to ensure that equations (14) and (15) shall hold for both arcs with the same values of the constants a and h ; and this condition ts satisfied by the figm-e. The relation expressed by 1 -j = 0, or 1 = 0, is in fact r r the same as that which must hold in order that A,' may vanish at A and D. We have supposed in the figure tTiree arcs, but it is obvious that the reasoning we have used will apply whatever may be the number of arcs; and as we may make this number as great as we please, we can finally obtain a curved line which difiers in length by as small a quantity as we please from the straight line ^5. Thus when the curvature is to have an assigned constant value, the solution will, as in the former case, coincide practically with the straight line which joins the two given points. 155. In the preceding four articles we have supposed that there is no restriction on the tangents at the extreme points. If the directions of the tangents at the extreme points are given, it will no longer be necessary, as in Art. 152, that V should DELAUNAY. 177 vanish at both limits. If the tangents at the extreme points are equally inclined to the straight line which joins the extreme points, and if also the two tangents and the straight line are in the same plane,^ then if the value of the curvature is not assigned, it will he possible to satisfy the conditions of the problem by a series of circular arcs as in Art. 154; and if the value of the curvature is assigned, it will be possible to satisfy the conditions of the problem by such an arc or such a series of arcs, if the distance of the extr^e points and the magnitude of the radius of curva- ture and the directions of the tangents have been given suitably, but not otherwise. But no solution has hitherto been given for the general problem when the directions of the tangents at the extreme points are assigned in a perfectly arbitrary manner. 156. We will shew that in certain cases a helix may be the solution of the problem. For suppose in equations (12) that a = 0, 5 = 0,/ = 0,/" = 0; assume x = hcoad, y = hsm6, z=Jc0; thus ^ = *J{h' + k'), and dd dx _ y dy _ ^ dz _ h d^x _ X d'^y _ y d'^z _ d^~~F+¥' W~~¥+¥' d?~ ' ^■= "'(1+4:) ="'{'+ ck ^{h'^-k') The first of equations (12) when these values are substituted becomes P'^ + V(FTF)} (^T^i = -^ <''^' and the other two become |^ + V(A^}(A^F)I" '••••''••■• ^^^^^^^ s from (19) we shall obtain k^/{h'+P)=c{h'-k') (20); P ^— ^/{je + ¥)\ (A^ + F)* And p = — 7 — ; thus from (19) we shall obtain 12 178 DELAUNAY. and from (18) J- T^- h^-W ^^^)' It appears from (20) and (21) that we cannot have A = Z;, and with this exception a helix may be a solution of the problem. 157. We will now return to Delaunay's notation. It may be shewn by performing some ordinary transformations t]jat the equa- tions (1) and (2) may be written thus : dy f 2m dx\ d f dx d^y\ _ ds \ p dsj ds \ " ds di) ' dz / 2m dx\ d f dx d^z\ _ ^ ds\ p dsj ds\ " ds ds^)~ These coincide with the second and third of equations (11) by supposing , ^ dx ^, ^ ^ 2m dx ^ oc=-5, ^ = -0, ^p^ = X,andl--^ = \, that is X = 1 5- . P And the equation (7) may be written , dx _ (. dy o d^ d^ ds ^ \ ds ds ds and thus the value of mp-^- coincides with that found for X' in Art. 152, by supposing 7 = — a. From equations (1), (2) and (7) in the form in which they are here given, we can deduce an equation coincident with the first of equations (11) ; so that the two solutions agree, as of course they should. Now in Art. 153 we obtained as the result of the symmetrical solution that in the case in which the tangents at the limiting points are unrestricted the required curve must be a plane curve, and that the plane of the curve must contain the line X _y _ z a h c' DELAUNAY. 179 If tiiis result be transformed into Delaunay's notation it leads to this conclusion ; the plane determined by z =a2/ + bx + c in his notation must contain the line determined by — = -^ = — ^ in ^ -7 -a -/3 his notation. For this to be the case we must have y = ^—7 — . This agrees with equation (6) and verifies Delaunay's conclusion ; but it is not obvious in what way he arrived at it. 158. It must be observed that Delaunay does not treat the integrated part of the variation as we have done in Art. 151 ; he considers that in virtue of his previous remarks we must always have m = at the limits of the integration. But if w = at the limits the curve is necessarily a plane curve, as appears in Art. 153 ; and this is obviously impossible when the tangents at the limits are so assigned that they do not lie in one plane. This furnishes additional evidence against Delaunay's views. Moreover this problem affords a good illustration of the re- marks made in the first part of Art. 146 ; for when the condition is given that the curvature of the required curve is to be constant the natural meaning of this condition would be that at every point of the curve up to and mcluding the limiting points the radius of curvature is to be constant. 159. The next problem considered is to find a surface of minimum area, the required surface being supposed to be bounded by a curve lying on a given surface. The problem had been con- sidered originally by Lagrange, in the case in which the bounding curve was supposed fixed; see Art. 18. Delaunay arrives at the known result that the required surface must be one that has at every point the sum of its two principal radii of cm-vature zero, Delaunay shews moreover from the equation which holds at the limits that the required surface must cut the given surface at right angles at every point of the curve of intersection of the two sm-- faces. Delaunay then generalizes his results by considering the multiple integral I ...\\ Idu ... dvdxdy f. \ 1 + du) /dzV /dzV fdz\ \dv) \dx) \dy) 12—2 180 DELAUNAY. 160. The next problem is to find tlie sm-face which has a given area and "bounds a maximum volume ; that is, z must be determined such a function of x and y as will make ^^dx dyz ^ maximum, while \\d^dy^[ '^Y fdz dxj \dy is to be constant. This problem was originally considered by Lagrange ; see Strauch, Vol. Ii. page 623. Delaunay obtains as the result that the required surface must be such that at every point the sum of its two principal curva- tures must be constant. He supposes that the required surface is to be bounded by a curve lying on a given surface, and he gives a geometrical interpretation of the equation which he finds must hold at the bounding curve. He then generalizes his results by taking the case in which \...\\\du.,.dvdxdyz is to be a maximum, while fdzV fdzV fdz^ \dv) \dx) \dy is to be constant. 161. Delaunay makes some further investigations respecting the surface which includes a maximum volume with a given area. He says that of all closed surfaces the sphere was known to be that which included the greatest volume under a given surface, but that this result had not yet been deduced from the equations furnished by the Calculus of Variations. He tried the question in another way, and although he did not succeed in arriving at a complete solution he gives his results. The problem considered is the following ; it is required to join by a surface of given area two curves of given length situated in two parallel planes, in such a manner that the included volume may be a maximum. The differential equations to which the problem leads are then given, and, assuming that the required surface will be a surface of revo- lution, it is proved that it must be a sphere. The problem is DELAUNAY. 181 given by Mr Jellett, witli some additional remarks on the last of the limiting equations; see his Calculus of Varmttons, pages 282—286. 162. The last example considered by Delaunay is the variation of the following expression which occurs in the theory of heat, ///^..«[(-)V (-)%(-)] .//...,.rV|i.(|)V(|)]. In concluding our account of Delaunay' s memoir it may be observed that the examples, although interesting in themselves, do not throw much light on the precise point which according to the announcement of the Academy of Sciences required illustra- tion, namely, the equations which must hold at the limits of the integrations ; see Art. 133. And there is very little that can be. considered as an application to triple integrals which was specially indicated. From the fact that the judges drew attention to Delau- nay's researches on the distinction of maxima and minima, it may be inferred that they, as well as Delaunay himself, were not aware that he had been anticipated by Brunacci on this point. CHAPTER VII. SAKEUS. 163. We Lave already stated that the prize offered Iby the Academy of Sciences of Paris for an essay on the Calculus of Vari- ations was awarded to M. Sarrus; see Art. 133. "We now proceed to give an account of the memoir which obtained the prize. This memoir is entitled Recherches sur le Calcul des Variations ; it is published in the tenth volume of the memoirs of the Savants Etrangers, and the date of publication is 1846. 164. The memoir consists of 127 quarto pages. It is divided into five chapters. The first chapter is chiefly occupied with formulae for differentiating integral expressions with respect to any parameter which they may involve ; the second chapter applies these formulae so as to obtain the variation of a multiple integral in an undeveloped form ; the third chapter developes this variation and shews how many equations must be satisfied in order that the variation may be zero ; the fourth chapter gives special develop- ment of the formulee in the case of triple integrals ; the fifth chapter applies the formulae to three examples. The memoir is extremly interesting and valuable, and contains a complete solution of the question proposed by the Academy. The formulse which are obtained are rather complicated, but this can hardly be avoided in the subject. The memoir is probably the most important original contribution to the Calculus of Variations which has been made during the present century. 165. The investigations of Sarrus apply to multiple integrals of any order, and some doubt has been felt with respect to the best method of giving an account of them. We shall confine ourselves to the case of a triple integral, because it appears that SARRUS. 183 no abridgement could render adequate justice to the general results given by Sarrus, and it would be almost impossible to comprise some of the more complicated formulse within the breadth of an octavo page. We may hope to succeed in giving an intelligible specimen of the investigations of Sarrus by taking the case of a triple integral; and we must refer the student who wishes to appreciate the full merit of the author to the original memoir. We shall not therefore give an analysis of the memoir article by article, nor shall we adopt the notation of the author. Sarrus uses the symbols x^^x^^x^^... for the independent variables; the lower limiting values of the variables are denoted by a single accent, as a?^', x^, x^, ... and the upper limiting values of the variables are denoted by two accents as x^' , x^', x^', ... We shall use X, y, z as independent variables, and shall denote as we have done heretofore the lower limiting values by the suffix and the upper limiting values by the suffix 1. The unavoidable complexity of the notation in the original memoir has led there to numerous misprints, which however are not of great importance. 166. We shall use then the following notation; by the ex- pression jdx jdy Jdzu we denote a triple integral ; we suppose that It is a function of the independent variables x, y, z, and of any dependent variable or variables, and differential coefficients with respect to x, y, and z. The integration in the triple integral is supposed to be performed, first with respect to z from the limit z^ to the limit z^^, next with respect to y from the limit y^ to the limit y^, lastly with respect to a? from the limit x^ to the limit ic^. It follows from the nature of definite integration that the limits z^ and z^ will not be functions of z, but may be functions of x and y ; the limits y^ and y^ will not be functions of y or z, but may be functions of x ; and the limits x^ and x^ will not be functions of x or y or z. The limits of the integrations being thus distinctly stated we shall not express them in our formula ; but they must always be understood. No confusion or difficulty will arise from our not ex- plicitly introducing the limits because we shall never have occasion to use any indefinite integral, and we shall not make any change in 184 SAREUS. the order of the integrations. Also when we have occasion to use a single or double integral involving respectively one or two of the variables a?, 3/, z^ we shall not express the limits, but they must be understood. This omission of the limits in our integrals may at first be a little perplexing to the student, but it is strongly recom- mended by the simplification thus effected in the formulse. 167. The symbol 7 will be employed in the following manner. Suppose u any fimction which involves a quantity ^ ; if in w we change ^ into q^ we obtain a result which we shall denote by 7 u. p This symbol is the one which Sarrus himself employs ; he calls it a sign of suhstitution. The use of this symbol will lead us to ex- pressions of the following forms, I I u, I I 1 u, lax I u, lax lay I u, ^ V X y ss J y J J ^ z ' 7 lat/u, 1 I jazu; these expressions do not require any explanation. This notation is certainly one of the great merits of the memoir, and in this respect nothing has probably been suggested which is of so much service to the Calculus of Variations as this sign of substitution since Lagrange introduced the symbol B. 168. We shall now shew how to differentiate an integral expression with respect to any parameter which it may involve; the formula is well known, but it may be interesting to see the method of Sarrus, and to exhibit the result by means of the symbol 7. Let F{t, x) denote any fimction of the parameter t, the variable X, and other variables if required. Put ^-^=n>-) (1), ^>=t(',-) (2); ; SARRUS. 185 "we have then identically d(f> {t, x) ^ dylr{t,x) .^. dx dt ^ '* Let Xq and x^ denote particular values of x ; then But from (2) and (3) by integrating with respect to x, we obtain Fit, X,) - Fit, X^) =jdx ^|r {t, X), {t,x,)-cf>it,x,)=jdx'^-t^^. Substitute these values in (4) ; we thus obtain dt fdxir it, X) ==jdx ^±^ + ^^^^t,X,)-^ir {t, X,) Now put u for yjr [t, x) for shortness ; then the last result may be expressed thus : d fj fjdu, dx.-^i dx.„^o where u may denote any function whatever. It will be observed that in accordance with the remark in Art. 166, the limits of the integrals are not expressed, but they must be understood. 169. We now proceed to diiferentiate a double integral with respect to any parameter which it may involve. We have seen that if u be any function whatever, d f y f J du dx, -^1 dx„ „^o dt J J dt dt "^ dt ^ ' 186 SAREUS. in this formula change u into \dyu', thus and as in the preceding article hence by suhstitution we obtain ^Jdxjdyu=^jdxjdy^+jdx^ny-jdx^fl\ •^-dj^^r^'-^^ry''' 170. We now proceed to differentiate a triple integral with respect to any parameter which it may involve. In the result of the preceding article change u into jdzu; thus -,- \dx\dy \dz u = Idxidy-j \dz u + ldx-^1 ' idzu- ldx~^l jdzu -1 \dy\dzu — -^7 Idyjdzu. dx^ J'l ^~dt Now transform the first term on the right-hand side by Art. 168 j thus . du ~dt ■^ Idx Idy jdz u = jdx Idy Idz +jdxjdy ^ 1^' u -jdxjdy -^ 1° u \dx-^l jdzu — jdx —■! jdzu ^ 7 idy I dz u — -y^° 7 Idy idz u. + dx, n^'l '^~dt SARRUS. 187 171. We may modify the form of the preceding result. It is evident that if t he independent of ^^, q q 1 ur=-rl u\ P P now Zg and z^ are independent of z, also j/^ and y^ are independent of y and z, and x^ and x^^ are independent of x, y, and z. There- fore we can alter the. order of some of the symbols which occur in the right-hand member of the result of the preceding article, and exhibit that result thus, du It jjdxjd^jdz u =jdxjd^jjdz I .jdxjd^fu^-jdxjd^fu^ -jdx fjdz u ^l -jdx fjdz u ^» + + 172. We will now give some formulae for differentiating quan- tities affected with the symbol 7 which will be useful hereafter. Let F{t, ^) denote any function of the parameter t, the variable f, and other variables if required. Let ^ (i, ^) denote the partial differential coefficient of F {f, ^) with respect to t, and i^ {t, f ) the partial differential coefficient of F{t, ^) with respect to ^; then we have now let u = F{t,x)', then thus we have d ^ _„^ du d^ r^ du dt X te dt dt X dx Suppose that ^ is independent of £c, then by Art. 171, d^ r^ du _ „5 du d^ dt X dx X dx dt ' 18S SAREUS. and finally d r^ r^ (du du d^ r u=i /du du d^\ (^\ \di dxdij ^ ^' we have dt ^ X \dt dx , =7 u—l u: and it m tins we replace u by 7 dy V y ^ -^ Now Idv T- = 7 \ — l "u; and if in this we replace w by 7 u. i ^ d,y y y ^ •« jd, d J. J/i J J/o I -J- I U= I I U— I I u. dy ^ y « y !s Hence by (1), r, J> (du , du d^ „yij> JfoX Idyl [-J- + -J- -j^]=l 1 u — 1 1 u; J ^ ss \dy dz dyj y « y » therefore r, Xdu r, Xdud^ JirX JfoJ . . idyl -j-=—\dyl i-^+7 7 u — J 7 u (2). J '^ dy ] ^ !^dz dy y a y ^ ^ ' In the applications we shall make of the formula (2) hereafter, ^ will denote either z^ or z^. 173. We shall now proceed to use the results already ob- tained in expressing the variation of a triple integral. Sarrus adopts an idea of a variation which had previously presented itself to Euler and Lagrange ; see Arts. 22 and 15. We consider then that we have a triple integral taken between limits for each of the three integrations; and we use the symbol u to denote the function to be integrated. Now u involves x, y, z, and any function v of x, y, z, together with the diflferential coefficient of v with respect to x, y, z', also u may involve any other function w of x, y^ z, together with the differential coefficients of w with respect to x,y,z', and so on. Now to obtain the varia- tion of the quantity which is denoted by any symbol, we suppose that such a symbol instead of representing a function of x, y, z, or of some of these variables, becomes a function of t also, where i is a new variable which is supposed to enter in a perfectly arbitrary manner; then if the quantity in question be supposed to be differentiated with respect to t, and t made equal to zero after the differentiation, the result is called the variation of that quantity. This idea of a variation had been used by Euler and SAERUS. 189 Lagrange as we have already intimated, and subsequently by Ohm ; see Arts. 22, 15 and 55. 174. In pages 45 — 47 of his memoir, Sarrus distinguishes be- tween two kinds of variations ; and we will now explain this distinction. Suppose we had such a triple integral as we have considered in the preceding article. We might conceive that the independent variables x, y, z received changes by variation as well as the dependent variables v, w, ... which occur in u. When however the integrations are taken between limits it is unnecessary to suppose that the independent variables themselves receive varia- tions; we obtain sufficient generality by ascribing variations to the dependent variables and to the limits of the integrations. When the variation of a function is taken on the supposition that the independent variables themselves do not receive variations, Sarrus calls the variation a variation tronquee^ and he denotes it thus, S. Then as he supposes his integrals to be taken between limits, he says that he is only concerned with these variations tronguees. 175. Now take the result obtained in Art. 171 ; then if we adopt the idea of a variation explained in Art. 173, and use the symbol 8 to denote a variation, we have the following formula : h \dx\dy \dzu=\dx\dy\dzhu + 1 ldyldzuSx^ — 1 jdyldzluBx^ + jdxl idzuSy^—jdxl jdzuSyQ ■+ Idx Idyl uSz^—jdxjdyl uBz^^. This gives in an undeveloped form the variation of a triple integral. 176. It is certainly not necessary to verify the preceding result, but it may be interesting to shew that it does agree with that 190 SAREUS. which had been given by previous writers ; this Sarnis does in the manner we will now indicate. According to Sarrus the known formula for the variation of a triple integral is the following : 8 \dx\dy \dzu= \dx\dy\dziZu — j- hx — -j- hy — -j- hz\ r, r, r, /duhx duhy duBz\ + jdx]dyjdz ^-^ + -^ + -^j . Sarrus calls this a known formula, but he does not say where it had been demonstrated. He probably had in view such a method as the following : B \dx\dy \dzu— 1 1 iBdxdydzu = ljjdxdydzBu+ in dSx dydzu 4- jljdx dBy dzu + jjjdxdy dBz u fffj 7 7 /<> du ^ du ^ du ^ duBx duBy duBz\ ^jjjdx dy dz\U -j.Jx-^^iy-.^Jz + -^+^ + -^y But it must be observed that before the researches of Poisson the variation even of a double integral had not been investigated, for the case in which the limits were variable, in an intelligible and satisfactory manner. The formula which Sarrus considers to be known will be seen to be analogous to that demonstrated by Poisson for a double integral j for Poisson's result is in which the quantity S F is what Sarrus would denote by ^ , dV ^ dV ^ ^^~l^^^~d[,^y'' see page 76. And the formula agrees also with the general result given by Ostrogradsky ; see Art. J. 2 7 at the end, where the quan- tity which Ostrogradsky denotes by Du is what Sarrus would de- note by ^„ dU ^ dU ^ dU ^ oU r- ox 7- mi ^— bz— ... ax dy dz SARRUS. 191 177. In order to shew that the formula given "by Sarrus agrees with the result which he considers known, we shall require some simple theorems of the Integral Calculus. Let u be any function ; then J az z « ' therefore \dx\dy \dz--T-= \dx\dyl u— \dx\dyl u (1). du dy Affain \dy-j- = / u — 1 u, ° J ^ dy y y ' change % into \dzu', thus \dy-j-\dzu = l jdzu — 1 jdzu; therefore jdy^jd.p^ + fu^-fu^l^ = 7 Idzu — l Idzu; yj yj therefore Idy Idz -j- = — jdy 1 u-^ + Idy 1 u -j^ + fjdzu-fjdzu (2); therefore jdx Idy \dz -7- = — Idxjdyl u-j^ + \dx Idyl u -^ + Idxl jdzu— jdxl jdzu (3). Again, hy (2) we have idx Idy -^ = — Idxl u -4^+ Idxl u-^ J J '^ dx J y dx J y dx *! r ^oC - -^^^^jdyu-l^^jdyw, 192 SAERUS. change u into jdsu; thus we shall obtain jdxjdyjd. £ = -jdxjd^fu ^ +jdxjd^ fu^ dx ■ Idxl \dz u-Y^ + Idxl \dzu J yj dx J yj + T^\dy^dzu-l^jdyjdzu (4). Now transform \dx\dy\dz —j — by means of (4), transform pccmy ItZs — ^r-^ by means of (3), and transform \dx\dy \dz —^ — by means of (1), and rearrange the results ; we thus obtain from the known formula of Art. 176, h jdxldy Idzu— jdxjdy idz (Bu — -j- ^x — -t- hy — -r- Bz\ + 7 Idy ldzuBx — 1 jdy idzuBx + jdx fjdzu (By - ^ Bx) - (dx fjdz u (.By - ^ B^ + jdxjdyl\(Bz-^By-^Bx) ^jdxjdyl\(Bz-^By-^Bx). Now it is obvious that Idxjdyl uBz —\dx\dyl uBz^, idxl ldsuBy= jdxl jdzuBy^, 1 jdy jdzuBx = 7 jdy jdzuBx^, SARRUS. 193 and similar equations hold when the suffix 1 is replaced hy the suffix 0. Hence we have S jdx \dy jdzu = Idx \dy \dz iZu — -T-Sx—-j~Sy—-^Sz] + 7 jdy j d.z iiSxj^ — 7 \dy \dz uhx^ ^fdxfjdzu(By^-'^Sx) -jdxfjdzu{Byyf^Bx) +jdxfdyl\(Sz,-^By-^Sx) -jdxfdyl\(Bz,-^Byy^Sx] Moreover Bz,. >- dz^ dx Bx = Bz^, ^.Vi- ax = hv Bx^ = Sx^, and similar equations hold when the suffix 1 is replaced by the suffix 0. Also t, du p> du ^ du -, X bu — J- ox — r- oy — 7- OS = ou. dx dy dz Thus the known formula of Art. 176 has been so transformed as to agree with the formula of Sarrus in Art, 175, 178. After obtaining the general expression given in Art. 175 for the variation of a triple integral, the next step is to shew how by integration by parts as many of the terms which occur in Bu. as possible are removed from under the signs of integration. This part of the subject is fully considered by Sarrus; we will give three of his formulae. 13 194 SAERUS. In equation (4) of Art. 177 change u into ud; thus \dx\dy\dzu--j-=— \dx\dy \dz-j- 6 + 1 \dy\dzu9 — 1 \dy\dzud - (dxf[dzu9^'+ fdxfidzud^' J yj ax J yj ax - jdxjdyfuep + [dxjdyl\e^ (1). In equation (3) of Art. 177 change u into u6; thus + jdxl \dzud — \dxl \dzu6 - jdxjdyfuep + jdxjdyl\ep (2). In equation (1) of Art. 177 change u into u6 ; thus Hdy\dzuf^-\dx\dy^dz^^J + ldx{dyl\e - {dx{dyl\e (3). 179. In his last chapter Sarrus applies his formulje to three examples. The first example is, to determine the surface which with a given area contains the greatest volume. The third example is, to determine the law of the density of a hody of given form and position in order that the integral \dx \dy \dz w -7—^ — 7- J J J dxdydz SAREUS. 195 taken throiigliout the body may "be a maximum or minimum, v being the density at the point {x, y, z) and w a given function of X, 1/, z, and v. The discussion of the first example is too long to be conve- niently given, and the third will find an appropriate place in the next chapter ; the second example we will now consider. We shall however in future omit the bar from the symbol 8. Sarrus has indeed said very little about what he calls a variation tronquee; see Art. 174. Perhaps this term and the corresponding symbol S were only introduced for the pm-pose of enabling him to compare his formula with the known expression for the variation of a multiple integral as in Art. 177 ; and no disadvantage would have arisen if the term and symbol had not been introduced into the memoir. 180. The following is the second example given by Sarrus ; to determine the law of density of a body of given form, position, and mass, in order that the integral /&/jy/&^|i + (|)+(^) + ^dy) \dzj taken throughout the body may be a minimum, v being the density at the point (cc, y, z). The mass of the body is equal to Idxldy Idzv, and since the mass is to be constant the variation of this expression must be zero. Moreover since the form and position of the body are known the variations of the limits Sx^, Sx^, By^, By^, Sz^, Bz^, are all zero ; thus the variation of the mass reduces to idxldy idz Bv, and this must consequently be zero. Again, put. fo.^{l + (|)V(|)V©]; then the . 13—2 196 SAKRUS. variation of the proposed integral is r , r , r , (l dv dBv , l ^^ dBv ."i-dv dSv J J J V ^^ ^^ r dy dy r dz dz) Then by the ordinary theory of relative maxima and minima we must have jdxldy jdz h ^ I dv dBv 1 dv dBv 1 dv dBv) _ r dx dx r dy dy r dz dz ' where c is some constant. Our object now is to transform this equation so as to reduce as much as possible the number of the signs of integration which occur with any term. We first transform jdxjdy jdz- -r- -r— by means of equation (1) of Art. 178 ; we obtain as the equivalent y 1 dv iff r dx — jdxldy jdz , Bv + 7 jdy jdz - -j-Bv — 1 \dti \dz - -^ Bv X J "^ J r dx ^ J "^ J r dx - (dx f f& i *^ & a. + Idx f [& i f^ f^ s» j y J r dx dx J y J r dx dx — Idxldy 1 — J- ~T^ ^'V + idxldy 1 - -^ -^ Bv. J J "^ z r dx dx J J '^ z r dx dx Next we transform \dx\dy\dz-~ -^ by means of equation (2) of Art. 178; we obtain as the equivalent — jdx jdy I -, \ dv a.- -^ - dy dz -, '' Bv dy + jdxl jdz - -j-Bv — idx 1 I dz - ^ Bv J y J r dy j y J r dy _ \dx \dy 7" i 1^ ^' «. + /■&• [dy I'l * ^« St,. i J ^ X r dy dy J j ^ z r dy dy SAERUS. 197 Lastly, we transform \dx\dy \dz - -,- -^ by means of equation (3) of Art. 178 ; we obtain as the equivalent 1 1 dv d. — T- — \dx \dy Ids -, Sv dz r _ y , j^\\ dv ^ r , i + \dx vdy 7 - -r- 3v — \dx \dy 1 - -j-8v. Substitute tbe equivalents thus obtained in the original equation ; the result will be ■J I dv T 1 dv ,1 dv\ f 7 r 7 ^^l A <^v dz, 1 dv dz, 1 dv\ ^ — Idxldyl (-j--j-^-\ 7--T^ 7-ov J i '^ z \r ax ax r ay ay r dz) 4- [fl {d 7^" /^i ^ ^^« "'" ^^ '^^° "^ ^"^ ^ j J ^ z \r dx dx r dy dy r dz) J yj \r dx dx r dy) J y J \r ax dx r dy) + 7 jc??/ \dz — =- Sy X ] ^ ] r dx — 1 Idii \dz - ^- 8v. X J "^ J T dx Hence we must have by the reasoning commonly used in the Calculus of Variations T 1 dv -J 1 dv T 1 dv r «a; y ay r dz _ dx dy dz ' this must hold for every point of the body. We have also certain limiting eqiiations, six in number, namely, 198 SAREUS. _*i /I dv dz^ 1 dv dz^ 1 dv\ _ js\r dx dx r dy dy r dz) ' Jli /I dv dy^ 1 dv y \r dx dx r dy = 0, rf^l_d^ _ ^ X r dx ' and three more wliicli can Ibe obtained from these Ibj replacing the suffix 1 Iby the suffix 0. 181. There are many misprints in the original memoir, as we have already remarked, hut they are not likely to give any trouble to a student except perhaps the following. In the third line from the bottom of page 119 are two mistakes; in the first term the dx" . factor -r-i- in the notation of Sarrus is omitted, and in the second term the factor —- is omitted. These lead to mistakes in those ax terms of Art. 155 which are numbered 9, 11, 15, 18, 21, 24; for -7^ is omitted in 9, 15, 18, and -y^ is omitted in 11, 21, 24. More- over in Art. 155 the terms numbered 14, 17, 20, 23 have the wrong signs prefixed ; and the terms numbered 30, 32 have in the notation of Sarrus w instead of -,— . dx^ 182. Some other remarks may be made for the use of the student of the original memoir. In his Art. 156 Sarrus interprets the equations which he has obtained in his solution of his third problem. Thus he finds that the equation dw d^v d^w _ dv dxdydz dxdydz must hold at every point of the body; and besides this certain limiting equations must hold. He appears to sum up his results at the bottom of his page 126 where he says, "by combining the different preceding conditions which hold at the limits we see that they reduce to this — -for all joints of the surface of the iody in question we must have w = 0." This must be understood to mean that at all points of the surface of the body w must vanish and so SAERUS. 199 must every differential coefficient of w witli respect to x, y, s, of any order. In other words lo must vanish identically at every point of the surface of the body. Moreover the result might perhaps he obtained more simply than in the way which Sarrus has adopted. For he obtains on page 123 as one of the limiting equations, an equation which ex- pressed in our notation is z\ 1 10 = 0. z The equation is to hold throughout what we may call one of the bounding faces of the body. Now if the equation just written be not an identity it really famishes an equation to this bounding face ; but the body is supposed to be given in form so that the equation to the bounding face is already known ; therefore the equation must be an identity. The only exception is that the equation 7 w = might happen to coincide with the known equation to the bounding surface ; but it may be shewn that this supposition is inadmissible by examining the equations from which 7 t« = was deduced. Again, in the method of Sarrus he might have observed that when some of his limiting equations are satisfied some other of these equations are necessarily satisfied also. Thus on his page 125 he has a sentence which in our notation will read thus ; moreover the fourteenth andffteenth terms loill give 7 7w7 = 0, 7 7-T-=0. y ^ y !s ax This is quite true, but it is not additional to what is already Zi known ; for he has already shewn that 7 w = 0, and therefore of 2/i •^i ^1 dto course 1 1 w must be =0, and he has also shewn that 7 -=- =0, y z z dx and therefore of course 7 7 -r- must be = 0. y z dx We may observe that the misprints which occur in Art. 155 of the memoir of Sarrus do not affect the validity of the inferences which he draws in his Art. 156. 200 SAERUS. 183. As a furtlier illustration of the method of Sarrus we will give in detail the investigation of the variation of a double integral, in which we will suppose that no differential coefficient of a higher order than the second occurs in the proposed expression. We shall require some formulae in the integral calculus which might be obtained from those we have already given, but for convenience we will investigate them here. It is obvious that /■ dx-Y-=l u~l u. (1) ; dx •'? change u into \dyu; thus \dx-^\dyu—l \dyu — l Idy that is \dx \\dy -^ + 1 'u-r^ — 1 \ -pV = 7 Idyu — l \dy u, J \J ^ dx y dx y dx) x] ^ x] ^ ' therefore Idxjdy -j-=l \dyu — l jdyu - Idxl 'u-^+ \dxl °u-^; J & dx J y dx change u into uB\ thus du dx \dx\dyu-T- = — Idxjdy + 7 jdyud — l \dyu9 -Idxfue'^^ + ldxf'ue'^ (2). J y dx J y dx ^ ' Again, in (1) change u into 7 u ; thus ' Idx -j-1 u=l 1 u — 1 1 u, i dx y «! y X y ' that IS, \dx/ -r--\-\dxl -f- -j^ = 1 i ti — I i u, J ydx J ydydx ^ y x y ' SAEEUS. 201 4.1, c fj n'^^^ -f'n^ Z^" L n'^udr] ; tnereiore \dxl -^- = 7 / u — l I u— lax I -7--^-; J p ax «! y ^ y ] y dydx change u into uO, thus \dxl u-^ = — Idxl -j-^ + 7 7 ud — 1 7 ud J y dx J ydx x y x y ^[g^f,,f'^_L^f^0^ (3). J y dy dx } y dy dx In the applications we shall have to make of this formula t] will be either ?/o or y^. Again, it is obvious that r, du „yi Jfo \a2/-j- = I u—l u; J '^ dy y y ' change u into u0, thus Hence jdxjdyu-T- = — jdxjdy-j- 6 + jdxl u9—\dxl ud...{^)i . and l^{dyu^ = -f\dy~e + ffue-l^fud (5). xj'^dy <«} dy ^ y oc y ^ J In the applications we shall have to make of the last formula, ^ will be either x^ or x^. 184. Let then C^i Cyi U= Vdxdy, J Xg J j/o where V is supposed a function of ds dz d^z d^z d'^z ^' ^' ' cfe ' dy^ 'ds? ' dxdy ' dy^ ' In this double integral we suppose that the integration is effected with respect to y first, and the limits y^ and y^^ may be fanctions of x. As the limits and the order of integration will 202 SARRUS. continue unchanged througliout the investigation, it will not be necessary to denote the limits explicitly, but they must be always understood. As in Art. 175 we shall have, using B instead of 8, BU=jdx[d^SV+ f^ldy VBx^-fidT/ VBx, +jdxfvBy,-jdxfvBy,. And ^rT_dV^ y dSz y dSz y d^hz ^ d^Bz „ d^Sz Tz ^^ '"'d^'^ "'^^ ''''Ibf^ '''•'dbUy^ "'"d^' where F^ denotes the differential coefficient of V with respect to dz -J- , and Vy denotes the differential coefficient of V with respect to ■J- , and Fj.^ that with respect to T-g , and so on. Thus \dx\dyhV consists of six terms, and all of these except the first may be developed by means of the formulse given in Art. 183. First; the term Idxldy -j- Bz does not admit of any transfor- mation. Secondly; by equation (2) of Art. 183, Thirdly ; by equation (4) of Art. 183, jdxjdyV,^=^-fdxjdy^Bz + jdxfVyBz-jdxfVyBz. SAERUS. 203 Fourthlj ; by equation (2) of Art. 183, J y '''' dx dx J y ""^ dx dx ' Out of tlie five terms on tlie right-hand side of this equation, the first and the last two admit of further transformation; by equation (2) of Art. 183, *'' J dx X J '^ dx + -[dxf^-^^Zz^^^[dxf^-^Zz J y ax dx J y dx by equation (3) of Art. 183, \dxf Vj^^ = - fdxf Sz4-(kM J y ^ dx dx J y c?a; V d«^J X y '^ dx X y ''' dx _ [dx f' — V (^\ - [dx f'Sz ^^(V ^V J"^^ y dy ^^Adx) r"^ 'y dx dyV'"' dx) ' a similar transformation can be made of [dx-f'V — ^» j y '"' dx dx' Thus we shall get on the whole jdx^dy K. ^ =jdxjd!, ^ S. X J ^ dx X j ^ dx 204 SAREUS. J & ax ax J y ax ax -H'4x{^'S-H>U^''fx It may be observed that as y^ and y^ are independent of y ^(V &^=:^i^Ii and— fF dy\^dy,dV^^ dy \ '"" dxj dx dy ' dy\ '"'^ e^a;/ dx dy ' Fifthly ; by equation (2) of Art. 183, \dx \dy F™ -^ — %- = — \dx\dy—Y^ -,— j J ^ ^y dxdy J J ^ dx dy *i r 7 ^^ dBz rj^o r d8z -^ir^-'d^-^.h^^^-d^ _ (dx fv—^+ (dx f" V — ^ ■ The first term on the right-hand side may be transformed by equation (4) of Art. 183, and the second and third terms may be transformed by equation (5) of Art. 183. Thus we shall get on the whole SARRUS. 205 J y ax J y ax ^ dV^,^ . .^0 f, ^F„, g^ -c/%f^^-c^. d^ X ] ^ dy + 7^' f V„ S. - 7^" /' V„ S. - f f r„ S. + f f F, 8. X y '^•' X y '^■' X y "^ x y '-' _ [dx 7^^ F — ^+ [dx f F — ^ -]dx ly V^ ^^ ^^ ^)^dx 1^ \^^ ^^ ^^ . Sixtlilj; Ibj equation (4) of Art. 183, [dxldv F — - - (dxldv^ — + jaxi^ 1..^^ jaxi^ ^vv dy ' The first term on tlie right-hand side may be transformed by equation (4) of Art. 183. Thus we shall get on the whole j dx^ dy Vyy ^ = yix^dy -J hz -{dxf^-^hz+\dxf^hz J y dy J y dy + }^^ly ^yy dy r'' 'y ^^^^ dy ' We must now collect the results obtained for the various terms occurring in \dx\dy hV. Thus on the whole we shall ob- tain the following as composing the value oi \dx\dyhV. First, a double integral, namely, {dxldy^^-^-^-^ + ^^+^^+^^ hz, J J "^ \dz dx dy dx? dx dy dy^ J 206 SAEEUS. Secondly, terms involving integration with respect to y only, nameljj Thirdly, terms involving integration with respect to x only, namely, ^ ^ /^Y _ ^y^j ^yyy'x dy \dxj dx ^l '"'Kdx) "^ dx ^ '' \ dy] Iz -Hl-'''t-^'- dV^dy^^d^fy dy, dx dx dx \ '^^ dx _j_ dV^ l^dy^ _ dV^ _ dVy,l ^^ + V f^V-F ^+F dy \dxj dx dy ' dSz) J dy)' Fourthly, terms involving no integral sign, namely, f: 1 1- f;. ^^ + fJ s. - 1 1 \-vj^+ vj sz X y dx X y dx Besides these there are in S U the four terms f'jdyVBx^ - f^jdyVBx^ + jdxl^ VBy^ -jdxl^ VSy,, SAERUS. 207 185. We have given in tlie preceding Article the develop- ment of the variation of a double integral; we, now proceed tq consider the relations which must be satisfied in order that the variation may vanish, supposing that no restriction exists with respect to the limits of the integrations. In order that the part of the variation which involves a double integral may vanish we must have dz dx dy dx^ dxdy dy^ ' this must hold for all values of x and y comprised within the limits of the integrations. Next, the term aflfected with the symbol 7 ( dy must vanish ; that is, when x = x^ must vanish for all values of y between y^ and y^ And since hz, -j^ , and hx^ are arbitrary we obtain the three equations F— — i — — ^ — F =0 F=0- ^'^ dx dy ~^' '^'^^ ^: y ^, these are to hold when x = Xj^ for all values of y between y^ and y^, so that we may conveniently express them thus, Xi / ^Y dV \ <^i •=''1 7Yf i^L-.^Uo, 7 F,, = 0, 7 F=0. •^ V dx dy J . ' ^ "^ ' ^ Similarly firom considering the terms affected with the symbol 7 1% we obtain Xq / fJY dV \ "^0 J'o ^^ j" Y,-^^-^^] = 0, 7 F,, = 0, 7 F=0. ^ V ^« dy J ' X ''=' ' X Next consider the terms affected with the symbols J ^a? 7^ and Idxl \ We shall obtain in a similar manner six equations, three 208 SAERUS. to hold when y=y^ for all values of x between x^ and a?^ and three to hold when y=yQ for all values of x between x^ and x^. We may write the first three thus, y\ ^ dx ^ dx dx dx\ '''' dxj dy \dxj dx dy ) ' 7 F=0. y The other three are obtained from these by changing y^ into y^. Lastly, the four terms without any integral sign must vanish ; thus we obtain four equations which must hold for special values of x and y, namely x = x^ and y = y^, and so on. These equa- tions are '^ y \ •"" dx ^V ' '^'vV'^'dx '^y ' '^v 'y \ ^-- dx "^J The equations we have obtained may of course be combined and thus simplified ; thus since we have already obtained f^ V^^ = and 7^" V^^ = 0, it follows that the last four equations reduce to 7 7 K„=0, 7 7 K„ = 0, X y " X y xy _^o „yi ^^ „^o J/o ^^ 7 7 F^ = 0, 7 7 7^ = 0, .1' . y X y " SAERUS. 209 186. "We may now make some comparison of the results of Article 184 with those obtained in Art. 143 by Delaunay's method. According to Delaunay's suppositions we have y\ — Vo when x — x^, and also when x = x^. In consequence of this a term affected with the symbol 7 / c?y vanishes because the limits of the integration with respect to y are equal when x^x^; similarly, a term affected with the symbol 1 I dy vanishes. Hence the variation of the double integral reduces to J J ^ \dz dx dy dx^ dxdy dy^ ) where ' ^ ^ dx dx dx dx\ '^'^ dx) ^cU^(dyX_dKy_dV^, dy \dxj dx dy ' ^' "'' \dx) "^ dx^ '"' and P^ and Q^ are formed from Pj and Q^ by changing the suffix 1 wherever it occurs into 0. This result coincides as it should do with that in Art. 143. 14 CHAPTER VIII. CAUCHY. 187. A MEMOIR by Canchy on the Calculus of Variations is published in the third volume of his Exercices d^analyse et de Physique Maihematique, 1844; it extends from page 50 to page 130 of the volume. This memoir may be described as a reproduction of a portion of the investigations of Sarrus with some difference of notation, and frequent reference is made to Sarrus throughout the memoir. In fact Cauchy himself does not appear to have considered his own memoir as more than a new exhibition of the method of Sarrus; thus he says at the end of his last chapter : The various formulse obtained in this last paragraph do not differ in substance from those obtained by M. Sarrus. They are however simplified by the nota- tion which we have employed.... Cauchy adds that he will develop the subject in some other memoirs and apply it to the solution of various problems. This design appears however not to have been accomplished. The memoir published by Cauchy may be considered an evi- dence of the favourable opinion he held of the method of Sarrus. 188. Cauchy's memoir begins with a few preliminary re- marks and is then arranged in nine sections under the following titles. 1. Definitions. Notation. 2. On the continuity of func- tions and of their variations. General properties of the variations of several variables or functions connected by known equations. 3. General formulge suitable for furnishing the variations of func- tions of one or more variables. 4. Properties of the variations of CAUCHY. 211 different orders. 5. On the variation of a simple or multiple definite integral. 6. On the different forms which may be given to the variation of a simple or multiple definite integral. 7. Com- parison of the formulse established in the fifth and sixth sec- tions. [The original memoir by mistake has third and fourth sections.] Differentiation" of a multiple integral relative to any variable different from those with respect to which the integrations are performed. 8. On the partial variation which for a simple or definite multiple integral corresponds to variations in the form of the functions which occur under the integral sign. 9. On the re- ductions which can be effected by integration by parts in the varia- tion of a simple or multiple definite integral. 189. The first four sections are very diffuse, but contain nothing new or important. In the fifth section a formula is obtained for the variation of a definite multiple integral which, as Cauchy remarks, is precisely the same as that obtained by Sarrus ; it is the formula which we have already given in the case of a triple integral ; see Art. 175. In his sixth section Cauchy gives an independent de- monstration of that formula for the variation of a multiple integral which, according to Sarrus, was known before he published his method; see Art. 176. We will exemplify Cauchy's demonstra- tion by applying it to the case of a triple integral. 190. We have to prove the following formula : h jdx \dy \dz u= \dx \dy j dz (Su — -j- 8x — -j- Si/ — -j- Szj f -, [ -, ( -, fduhx duhi duBz\ By \dx\dy\dzu, as formerly explained, we understand a triple integral in which we have first to integrate with respect to z from z^ to z^, then with respect to y from y^ to y^, and lastly with respect to X from x^ to ic^. Now let X=x + hx, Y=y + By, Z-z + Zz, where hx, By, Bz are indefinitely small arbitrary functions of x, y, z. Let U denote 14—2 212 CAUCHY. what u becomes when x, y, z^ are changed info X, Y, Z^ respec- tively, and also any function of ic, 3/, z^ involved in m, receives an indefinitely small arbitrary increment. Then the varied value of the triple integral is jdxjdYJdZU; the limits will be found by keeping the same limiting values as be- fore for X, y, z. It is important to observe that since Bx, By, Bz are quite arbitrary we can obtain all the necessary generality in the varied value of the triple integral by retaining the original limiting values of x, y, and z. The variation of the triple integral will be found by subtracting the original value from the varied value. Now it is obvious that the complete variation will be obtained by determining separately the parts of the variation which arise from the change of w, x, y, z into U, X, Y, Z, respectively ; and when we are considering the change of one of the quantities the others may be supposed to retain their original values ; the terms thus neglected are in fact of a higher order than those which are retained. Thus by putting u + Bii for U we find that the term arisino^ from the variation of u is dxjdy \ dz Bu. IsTow consider the term which arises from the change of z, into Z while the other quantities retain their original values. The integra- tion with respect to z is the Jirst pe7'formed; hence by the change of z into Z the triple integral is changed into jdxjdyjdzu^^, 7 rjr where the limits of x, y, z are the orioinal limits ; and -j- is the dz differential coefficient of ^ with respect to z only, that is, supposing X and y constant. Now dZ d , -> . , dBz CAUCHY. 213 thus the term in the variation of the triple integral which arises from the change of z into Z is dZz \dx \dy ids u -5— . The simplicity of this process arises from the fact that the integra- tion with respect to z is the first performed. Now let us consider the part of the variation of the triple in- tegral which arises from the change of y into Y. We maj conceive that the order of integration in the original integral is changed so that y is now the variahle with respect to which the first integi'ation is performed ; we know from the Integral Calculus that this can always be done by making suitable changes in the limits of the integi'ations. Then as before we shall find that the term in the variation of the triple integral which arises from the change oi y into Fis dy the limits being as we have already intimated adjusted to the new order of integration. Now restore the original order of integration ; then we obtain for the required term / this result obviously coincides with that which was proposed to be proved. 214 CAUCHY. This investigation seems more simple than that given by Ostrogradsky ; see Arts. 327 and 128. The simplification arises from considering the variables separately instead of simultaneously. 191. In his seventh section Cauchy shews that the form given for the variation of a multiple integral by Sarrus coincides with the form known before ; this Cauchy does in the same way as Sarrus ; see Art. 176. Cauchy proposes a new notation which he strongly recommendSi According to the notation of Sarrus we have r*i , du -^1 3 I dx-T- = I u—1 u: J^o dx X X ' Cauchy proposes to express this result thus r^i , du ^=^1 I dx -j-= \ u. Similarly what Sarrus would express thus 7 7 u — 1 7 u—i 1 u + l 7 u, X y X y X y an y Cauchy proposes to express thus _ x=x-^ y=y^ I I ^' x=Xo y=yo This latter is the form that Cauchy really uses ; but apparently by a misprint he gives on his page 100 such a symbol as the following *i Pi I 1^- This however seems more convenient than the symbol which Cauchy really uses; and it may be rendered still more commodious by writing it thus u. To exemplify the use of this notation we may take the general formula for the variation of a double integral which has been given + L' + + CAUCHY. 215 in Art. 184 ; that formula expressed by tlie aid of the new notation wUl stand thus, J J \dz dx dy da? dx dy dy^ ) dy \dx) dx dy J where it must be observed that -j- will stand either for -^ or for 7 dx dx —^ as may be required. Cauchy says on his last page that on speaking to Sarrus respect- ing this new notation by means of which the difference between two values of a variable is expressed by a single symbol he learned that the same idea had occurred to Sarrus himself. Perhaps w^ may infer that Sarrus on trial was not satisfied with it as he did not use it in his memoir. ^=$ Cauchy also proposes to use | u in order to denote what Sarrus denotes by 7 u; we shall not follow Cauchy on this point but adhere to the notation of Sarrus. Thus we only use Cauchy's notation when we have to express the difference between two values of a variable, and we use Sarrus's to express a single value of a variable ; there is then no possibility of confusion. 216 CAUCHY. 192. Cauchy's eighth section contains nothing new or impor- tant. In his ninth section he considers the transformations which are to be made of the expressions by means of integration by parts. This section is illustrated by an example which we will now give in detail. Let u be an unknown function of x, y, z] let ■« be a function of X, y, z and , , . Let hs denote that part of the variation of dtC cty ci/Zf the triple integral I c?ic I d'y I c?5;v which arises from the variation of u ; then we have ^'^hhh'^dfdz (^)' d% where r is the differential coefficient of v with respect to ^j — j—y- . '■ dx dy dz The limits of the integrations are supposed to be denoted as hereto- fore. We propose then to reduce '^s by means of integration by parts. .^^ , d^hu _ d d^Bu dr d^Bu dx dy dz dx dy dz dx dy dz'' thus (1) becomes §5 = \dx\dy \dz -^r-^ — 5 — \dx\dy \dz-^- ^ — 5- (2). j J ^ J dx dydz J J ^ J dx dy dz ^ ^ By equation (4) of Art. 177, [j (j fj d d'^Zu 1^1 r, r, d^hu Idxidy \az-j- r^ — 7-= \dv\dzr-^ — r j J ^J dx dydz ^^oj ^ } dydz J y J dy dz dx J y J dydz ax _ U U f .#^ ^ + \dx kyl'r p^^^. J J .'' ^ dy dz dx J J '^ ^ dy dz dx CAUCHY. 217 Thus (2) becomes J y J ay dz ax J y j ay az ax -{d^ldyiy II % +l,.j,yfr H '^ (3). J J '^ tvU Ct'JV tltl/ J J ^y tX'iV UjkK/ It will be observed that Zu is not differentiated with respect to x in any term of (3). In equation (2) of Art. 177 change u into r -r— ; thus r, [, d^Bu fj (-, dr dhu rr''d^z=-rrd-y^ _ [^yfr — —'+ (dvfr—'^ J ^ z dz dy ] ^ z dz dy + f ' \dz r dhu dz Thus the first term on the right-hand side of (3) becomes ,^1 ^yl f -, dBu \dzr-r- '•i-oj y 2 dz dy '■3?oj ^ z^ dz Hy i"*! f , ( 1 dr dSu Similarly the second term on the right-hand side of (3) be- comes f ,2'i /*- dr dSu \dx\ \dz ^- J 'yoj dx dz f , ( T „^\dr dBu dz, T , T , ,^0 dr dBu dz^ dy J J ^ '^ dx dz dy d^r dBu [^ /*7 [^ dr dBu -\d^\dy]dz-y-j-~^ J J J UiX U/ll (Jjii 218 CAUCHY. The third and fourth terms on the right-hand side of (3) we shall not transform ; we proceed to the fifth term. 75> 7 In equation (2) of Art. 172 change u into r -j— -^ , and put sr^ for ^; then observing that z^ is independent of z we obtain the following result, r, -^1 d'^^u dz^_ r, rj^id^u d dz^ j ^ « dydzdx~ ] ^ ^^ dz dy dx .2'i„*i dhudz^ r, r^\dz^dz^ d d^u , . 'yo « dz dx ] " ^ dx dy dz dz ^ '' Now by equation (1) of Art. 172 „*i d dz^ _ d _*i dzj^ rfi dz^ dz^ dr « dy dx dy !^ dx * dx dy dz ' Thus a part of the first term on the right-hand side of (4) can- cels a part of the third term, and we obtain r , „^i d'^tiu dz^ _ r , -^1 dhu d rf^ dz^ J ^ « dydzdx~ ] "^ ^^ dz dy « dx y\ J'l dBu dz^ ^Vo « "dz dx /*, „*i d'^hu dz^ dz^ J ^ « dz^ dx dy' A similar formula holds when z^ is changed into z^. Thus we obtain hs dBu dz _ C [dvf r ^ ^ + r^ U 7'V — — - 'ifoj "^ « dz dy *xoJ •^ « dz dy dy '■^oj ^ X dz dy dr dSii dy dz .^1 [ -, f , dr dhu CAUCHY. 219 r 7 Vi C T dr dou — \ax\ \dz -^ 7- j Voj ax dz + (dxldvf- ^^-Idxldvl'"———' j } ^ « dx dz dy J ] ^ ^ dx dz dy fy r? Tt ^^^^ dSu + jaxjdy \dz dxdy dz J j y j dydzdxj y j dy dz dx + (dJdyf'^-p^^fr't^- (dx(dyf° '^pj.f'r^ J ] ^ '^ dz dy ^ dx J J ^ ^ dz dy « dx -(dxrf'r^-pp+fdxf'fr^^ J 1/0 •» dz dx J 'vo « dz dx , fj (j 7^' ^^^'"' ^^1 ^^1 (j fj n'^" d^Su dz„ dz. + \dx\dyl r-^-Y--T^ -T^— IdxldyJ r ^-^ -r-S- -—? . J J ^ « dz^ dx dy J j ^ ^ dz^ dx dy In some of these lines terms occur involving integrations witli respect to z and differentiation of Su with respect to z ; such terms admit of further reduction. In the first line we have Idz ^—r-, and ldzr-^— = \ rBu— Idz-rrSu. J dz '^0 j dz In the third line we have \dz -^ — ^— , and J dy dz /*, dr dBu .'^idr^ /", d^r ^ Idz -r: =— = -Y-bu— Idz -y — =- bu. J dy dz ^0 dy J dydz In the fourth line we have Idz -^ =— , and J dx dz /*, dr dBu .'^i dr ^ f, d^r . dz -^ 7— = -y- bu — Idz -J — Y ou. J dx dz «Q dx J dxdz 220 €AUCHY. In the sixth line we have ds -^ — ^ — 7— , and J ax ay az d^r dBu ,^1 d^r » f^ d\^ r, d'r dlu ,^1 ^V . r, I as -7 — 7 7— = -^ — 7- ow — I as j tticaw az ^'^odxdy J Su. dxdy dz ^stodxdy j " dxdydz In the seventh line we have \dz r -^ — j- -f^ and \dz r -^ — ^ -^ , j ay az ax J dy dz dx , r 7 d^Su dy^ _ 1*1 dy^ dhu T , d^u dr dy^ j dy dz dx ^<^o dx dy J dy dz dx ' - r , d^Bu dy^ _ 1^1 dy^ dSio T , dBu dr dy^ J dy dz dx -''o dx dy J dy dz dx ' Thus finally Bs = Cf'CrBu-Cf'[dzfBu -f'idvf'r — '^ + C (dvf\^—^ '^oj ^ dz dy '■^oj "^ ^ c?s c?y ,^1 r , ,«i c?r ^ 1^1 r 7 r 7 ^V r. — Idyl -T-ou + \ \dy \dz -^ — =- bu '^0 J ^ ^0 ay '-^0 j "^ j dy dz — \dx\ I -7- Sit + ma; I \dz -^ — r Sw j '^0 «oaa; J 2/0 j dxdz + (dxidv f"- ——'-IdxUv l'"t^^ j J ^ ^ dx dz dy J J ^ z dx dz dy + dx dy L ^^ Bu- dxfdy (dz . J. Bu J !/ 1^0 dx dy J y ] dy dz dx + [dxf' f'rp^-idxf' [dzf^±^ J y xo dx dy J y ] dy dz dx + (dx (dy 7" f? # 7^' ,• ^ - (dx (dy f'^^ f , ^ J i ^ X dz dy z dx J J ^ « dz dy « dx — I dx I CAUCHY. 221 yi/i dZudz. [-, ,i/i„^o dBudz„ 7 r -J- -j^+ dx\ 1 r -J- -y-° Vo « az dx J Vo !« dz dx d^Su dz, dz, f , r , „^o d'^tiu dz^ dz +hh '»' '■ ^ £ I* -M'^^^ '^ '• ■«f dz'^ dx dy J J ^ x: dz^ dx dy ' 193. As a particular case of tlie preceding result, Cauchy supposes that r = 1. Thus we obtain ^5 = jdxjdy jdz dx dy dz ■^1 ?/i ^1 = 1 11 Bic ^0 Vo Xq _ /^'i r^ ^'^^dZudz^ ^x ^ ^ f - „yi ,^1 dy, dhu r, ^yo ^1 dy^ dSu — dxi -p--, -\- \dxl -p -^— J y '-^0 dx dy ] y ssq dx dy *^i c?Sm c?^2;, r -, r , „*o dl>u d^z^ +W'^^Cws*/-M'^^' ^ dz dxdy j j ^ dz dxdy J % IS dz dx J Vo yo ^ dz dx ^1 J^Sm c?^, dz. r , r , _^o £^^§2* dz^ dz^ ■^hh'^l^^^-^-hh^ « c?^^ dx dy J J ^ z dz^ dx dy ' Of the eleven terms here given Cauchy has omitted the sixth and seventh. With this particular case Cauchy's memoir terminates. 194. We can now conveniently introduce the third example which Sarrus gives in illustration of his formulse; see Art. 179. The example is the following ; to determine the law of the density of a body of given form and position in order that the integral dx dyjdzw^^^ 222 CAUCHY. taken throughout the Ibody may be a maximum or a minimum, V being the density at the point {x, y, z), and w a given function of X, y, z and v. Since the form and position of the body are given there are no terms in the variation of the triple integral arising from the vari- ation of the limits. Thus we have for the variation of the proposed triple integral \dx \dy \dz -^ — ^ — ^hw+ \dx \dii \dzw-r: — 5; — ^ . J J J dxdydz J J J axdydz The first of these two terms is equal to \dx Idy 1< , d V dw ^ az -^ — = — =- -y- 0?; ; dxdydz do the second developes into the twenty-two terms given as the result of Art. 192 provided in that result we change r into w and u into v. Of these twenty-two terms the twelfth is the only term which involves a triple integral ; this must be united with the term just given, so that we obtain \dx Idy , / d^v dw d^w \ -J dxdydz dv dxdydz J This must vanish by the ordinary principles of the Calculus of Variations; hence d% dw d^w _ dxdydz dv dxdydz This partial differential equation must be solved to find v, the solution of course involving arbitrary functions ; and the arbitrary functions must be determined from the limiting equations which we shall now examine. We may consider the given body to be bounded by six faces. Two of these six faces we will call the upper and lower ; they are determined by the known values of z^ and z^ in terms of x and y, and may thus be of any form. Two of the six faces we will call \hQ front and hack ; they are determined by the known values of y^ and y^ in terms of x, and are therefore cylindrical having their generating lines parallel to the axis of z. The remaining two CAUCHY. 223 faces we will call the right and left; tliey are determined by the known values oix^ and x^, and are therefore planes perpendicular to the axis of x. In particular cases these six faces would assume particular forms. For example, suppose the given body to be the ellipsoid determined by the equation x^ iP' ^ , I- •?- -^ — = 1 • the upper and lower boundaries are the portions of the surface of the ellipsoid determined respectively by \\\Q, front and hack boundaries are the portions of the ellipse in the plane of (a;, y) determined respectively by the right and lefi boundaries are the points on the axis of x for which x = a and x = — a respectively. Thus in this example two of the six faces degenerate into curves and two into points. The eleventh term in the result of Art. 192 gives \dx\dy I ^ — ^ hv ; j J ^ ^ssodxdy ' and in order that this may vanish, since hv is arbitrary, we must have /i dho /o d''w - 7 , , = 0, and 1 -j — r- = 0. ^ dxdg s dxdy The ninth and seventeenth terms in the result of Art. 192 must be united because they involve the same arbitrary term; thus we get Uxldvl'^i^—'^ — l'w^-^' j J ^ !^ dz \dx dy dy « dxj ' 224 CAUCHY. and in order that this may vanish, since —7— is arbitrary, we must have -,^1 [dw dz, d „^i dz „^i fdw dz^ d „^i dz\ _ ^ \dx dy dy '^ dx) ' ^ \dx dy dy or as we may write it _^i (dto dz^ dv) dz^ dw dz^ dz^ d^z^ \ —c\ « \dx dy dy dx dz dy dx dxdy) The tenth and eighteenth terms in the result of Art. 192 lead to a similar equation with z^ in the place of z^ . The twenty-first and twenty-second terms in the result of Art. 192 lead respectively to iz ax ay « ax dy We have thus proved that the following equations must hold, " dxdy '' -*i (dvo dz^ dw dz^ dw dz^ dz^ d^z^ \ _ « \dx dy dy dx dz dx dy dxdy) ' /i dz^ dz^ m dx dy The last of these gives 7 w = 0, for -^ and — are inde- ^ z ax dy pendent of z. The equation 7 ?« = shews that w must vanish identically for all points of the upper boundary. For if w does not vanish iden- tically w = must coincide with the known equation to the upper boundary ; and then we shall not have the other two of the above three equations satisfied. For fi'om 7^ ?o = we obtain by difierentiation ^i fdio dw dz\ _ J 7*' /^ , ^ ^■^A _ Q . z \d^ 'dz dx) "" ■ ' z \dy dz dy) ~ ' CAUCHY. 225 ^nd from these combined with the second of the above three equations we deduce ss dx ' !^ dy~ '' s dz~ '' and these could not be true if t(; = were the equation to a surface ; dw dw -. dw 1 . 1 1 • 1 /. because -r- , -7- , and -j- can only simultaneously vanish for special points on a surface and not for any continuous portion of a surface. Thus w must vanish identically for all points of the upper boundary. And similarly w must vanish identically for all points of the lower boundary. The eighth term in the result of Art. 192 gives \dx\ idzSv-i — r. J Voj dxdz This involves the value of Sv for the front and back of. the given body. Confining ourselves to the former, we obtain J/i d\o y dx dz The fourteenth term in the result of Art. 192 gives hxfbz—^ — ' J 1/ J dy dx dz and from this we obtain „yi dw _ y dz~ ^ since the factor -^ is independent of y. From the last two results we infer that w must vanish iden- tically for all points of the front boundary. For if -y- be denoted by w we have 7 w; = 0, and / -7- = ; y ' y dx 15 226 CAUCHY. and the first of these equations shews that w must vanish identi- cal! j for all points of the front boundary, or else w' = must coin- cide with the known equation to this front boundary ; but the latter supposition is inconsistent with the second of the above two equa- tions. Thus w must vanish identically for all points of the front boundary, and from this and the fact that w vanishes identically for the points common to the front boundary and the upper boundary, we infer that w must vanish identically for all points of the front boundary. Similarly from the remaining part of the eighth term and the sixteenth term in the result of Art. 192, we conclude that w must vanish identically at every point of the back boundary. From the sixth term in the result of Art. 192 we obtain 1 -^ — T- = 0, and 7 -7 — 7- = ; "c dy dz X dy dz and fi:om these terms combined with what we already know respect- ing w we conclude that w must vanish identically at every point of the right and left boundaries of the body. Thus we conclude from the terms that we have examined, that \o must vanish identically at every point of all the bounding faces of the given body ; and supposing this to be the case we shall find that the remaining terms in the result of Art. 192 vanish. 195. We will close this part of the subject by giving the com- plete development of the variation of a triple integral in the case in which no difierential coefiicient of a higher order than the first occurs in the proposed expression. Let then 1 1 1 Vdxdydz denote the proposed ti-iple integral, where „.„.-, du du du F is a lunction 01 x,y,z,u, -^-^ ^ , -j- . The integration is supposed to be efiipcted first with respect to z from z^ to z^ , then with respect to y from y^^ to y^ , and then with respect to x from o^^ to a-j. • CAUGHT. 227 Let the differential coefficient of V with regard to -j- he de- noted hj X, the differential coefficient of F with regard to ~ by Y, and the differential coefficient of Fwith regard to ~ by Z. Thus ^T^ dV ^ , „dBu . -^-rdSu rydtlU oV=--j-du + X-j h Y-J-+Z-J—. au ax ay az By Art. 175 we have in the notation of the present chapter S I M Vdx dydz— \\\B Vdx dy dz + ^^[dyldz VBx +jdx Q jdz VSy + jdxjdy Q VSz. There are four terms in B V giving rise to four terms in IJJBVdxdydz. The first term is not susceptible of transformation. The second term is to be transformed by equation (1) of Art. 178 ; this gives jdxldy Idz X-j — =— jdxjdy jdz -j—Bu The third term arising from 8 F is to be transformed by equation (2) of Art. 178 ; this gives jdxjdy jdz Y—j— = — jdxjdy jdz—j- Bu + \dx\ Idz YBu— \dx \dy \ Y-j^ Bu. J '2'oj j J --' ^0 dy 15—2 228 CAUCHY. The fourth term arising from S F is to be transformed by equa- tion (3) of Art. 178 ; this gives 4- \dx\dy\ ZBu. Thus we have finally BJjjvdxdyd.=jfjSu(^^-'^^y^-^)dxdydz + \2fdyjdziVBx + XSu) + jdxMdz (vBy - X^Ju+ YBu^ + jdxjdyfUvBz -X^Bu-Y^Bu + ZBuy Of these terms those affected with the symbols Cj^yj^^ and jdxQdz vanish when we confine ourselves to the particular case considered by Delaunay, as we have already explained in Art, 186. The re- maining terms agree as far as they go with the result given in Art. 144. CHAPTER IX. LEGENDRE, BEUNACCI, JACOBI. 196. We are now about to give the history of that part of our subject which relates to the criteria for distinguishing a maximum from a minimum, and for ascertaining when neither a maximum nor a minimum exists. We have ah'eady intimated in Art. 5, that Legendre had arrived at some results on these points, and that Lagrange had shewn that further investigations were required in order to ensure the accuracy of Legendre's conclusions. The requisite investiga- tions were supplied by Jacobi in 1837, and the memoir which Jacobi then published has given rise to an extensive series of commentaries and developments. Before however we proceed to Jacobi's investigations, we will give an analysis of Legendre's memoir and of some others connected with it. 197. Legendre's memoir is entitled Memoire sur la maniere de distinguer les maxima des minima dans le Calcul des Variations. It is printed in the volume for 1786 of the Histoire de VAcademie Royale des Sciences; this volume is dated 1788. The memoir ex- tends from page 7 to page 37. There is an Addition to the memoir on pages 348 — 351 of the volume for 1787 of the Histoire ... ; this volume is dated 1789. 198. The first investigation in Legendre's memoir is in sub- stance the same as that which we have given in Art. 5. He shews 230 LEGENDRE, BEUNACCI, JACOBI. by the method there used that the problem he is considering may be reduced to the investigation of the sign of In Article 5, we supposed Zy^ and Zy^ to be zero, so that the part of the above expression free from the integral sign vanishes. Legendre adopts the following method with respect to the inte- grated part of the above expression; the value of X is to be determined from a differential equation and it will therefore in- volve an arbitrary constant, and this arbitrary constant may be supposed to be so taken as to make \ (%i)^ — \ (^3/0)^ vanish or have the same sign as the part of the expression under the in- tegral sign. 199. Legendre next considers the case in which the integral of an expression /(cc, 3/,^, q) is to be a maximum or a minimum, where p = ^ and q^ = -~ . The investigation is similar to that already given, and the conclusion is that the result found by the ordinary processes of the Calculus of Variations will be a maxi- mum if ~-^ is always negative between the limits of the integra- tion, and a minimum if it is always positive. Legendre then says that it is easy to generalise these results and to infer that the ordinary processes will give a maximum, if the second differential coefficient of /(ic, y, -j- , -r^, ... ] with re- spect to the highest of the quantities -^ , -fi,--- which it in- volves is always negative between the limits of the integration, and a minimum if that second differential coefficient is always positive. 200. Legendre next considers the case in which we have to find the maximum or minimum of [/[x] y, jj) dx, supposing that X is susceptible of variation as well as y. The investigation is. LEGENDEE. 231 now more complicated than that in Articles 5 and 198 ; the re- sult however is the same, namely, that there is a maximum or mmimum according as -j4 is constantly negative or constantly positive between the limits of the integration. 201. Legendre next supposes that we have to find the maxi- mum or minimum of \f{x, y,p, ) dx, where (/> is to he determined from the differential equation -^ = -v/r, in which i/r is a known function of x, y, p, and ^. The result at which Legendre arrives is wrong, and the correct result was afterwards given by Brunacci. 202. Legendre then illustrates his investigations by some ex- amples. He first considers the case of the solid of least resistance, and he shews that the ordinary result is not necessarily a minimum. He then considers the problem in which among all curves of given length having their extremities in two fixed points, that is re- quired which has its centre of gravity lowest; here his method indicates that the catenary does possess the required property. Then he considers the problem in which a curve of given length is to be drawn between two fixed points, so that the area bounded by the curve, the ordinates of the fixed points, and the axis of abscissas shall be a maximum or a minimum. He shews that the required curve will in some cases be a circular arc, and in other cases will be composed of a circular arc and one or two straight lines ; we shall have occasion to return to this point hereafter. The three examples thus discussed by Legendre form a very interesting and instructive part of his memoir. Finally Legendre takes the problem of the brachistochrone in which the moving particle is to pass firom one given curve to another, starting with an assigned velocity. Then the expression to be made a minimum is V(l +/) dx '^{y-c + k)' 232 LEGENDRE, BRUNACCI, JACOBI. where h is tlie heiglit due to tlie assigned initial velocity, and c is the ordinate of the point at which the motion begins. Legendre takes c and x to be susceptible of variation, as well as y and p, and by a laborious investigation he arrives at the result that the time of motion is necessarily a minimum if the curve described be a cycloid, which meets the two given curves in points where the tangents to those curves are parallel, and which cuts the lower curve at right angles. 203. In the addition to his memoir, Legendre makes some remarks in order to strengthen two points in his conclusions. He says that he has to shew in the first place that the quantities which he supposes determined by differential equations, like the \ of Article 5, are necessarily real ; and in the second place that, as we have stated in Art. 198, the arbitrary constants which occur in the solutions of the differential equations can be chosen so as to make the integrated part of the terms of the second order in the variation zero, or of the same sign as the unintegrated part. Accordingly he makes some observations in order to establish these two points. 204. Two remarks may be introduced here. In the first place it must be remembered that all Legendre' s investigations are subject to the objection indicated by Lagrange ; see Articles 5 and 6. Legendre does not solve the difi"erential equations which he obtains, so that there is no security that the quantities he uses retain always finite values ; and Lagrange shewed that in a simple example Legendre' s conclusions were not necessarily true. In the second place, in all investigations with the view of distinguishing maxima from minima values, it is of course necessary that we should retain all the terms of the second order which can occur in our expressions. Now such formulfe as those of Poisson and Ostrogradsky in Articles 102 and 124 are only true to the first order, and consequently cannot be used in any investigation in which we are discriminating between maxima and minima values. This is one of the reasons which render it advisable to avoid giving a variation to the independent variable; see Art. 25. If, for example, we vary y and not x, then we have hp absolutely the BRUNACCI. 233 same thing as -j^ . If however we vary both y and a?, it is shewn in elementary works, as in Art 39, that 5, _ d^y dhx ^ ^ dx ^ dx ' this equation however is not accurately true, but only true to the first order. For dy = pdx, dy + dBy = (j) + Sp) {dx + dSx) ; therefore ? _ ^^y —pdhx _ /dSy pdSx\ f dBx\~^ " dx + dSx \dx dx J \ dx J ' thus in order to be true to the secoiid order we must take ^ _ f^^^ pd8x\ / dSx\ -'- \dx dx J \ dx) "^ and in fact Legendre uses this value of Bp on page 15 of his memoir. 205. We have next to consider two memoirs by Brunacci. The first of these is entitled, On the criteria which distinguish maxima from minima in integral expressions ; it is published in the Memorie delVIstituto Nazioncde Italiano, Vol. I. part 2. Bologna, 1806. The memoir extends over pages 191 — 202 of the volume; its object is to correct an error in the memoir which Legendre published in the History of the French Academy for 1786 ; see Art. 201. Suppose we have the integral I Vdx where V involves x, y, -^ , and z, and z is determined by the differential equation -^--Z^ where Z is a, function of x, y, -^^ and z. Then Legendre arrives at the following result ; Vdx is rendered a maximum or minimum by the ordinary processes according as d^V . -y-j is constantly negative or constantly positive between the 234 LEGENDEE, BEUNACCI, JACOBI. limits of the integration, wliere p stands for -^. Legendre's metliod is rather obscure and Brunacci follows it; we will here give the investigation in the usual manner, and we shall obtain the same result as Brunacci. 206. Let \ denote a function of x at present undetermined; then we may consider that we have to find the maximum or minimum of and we will denote this expression by U. Now, considering only terms of the first order, we have ^V=^hz-\-^hy + ^hp=^Ahz + Bhy + Chp say; hZ^^Jz^ — hy + '^hp^Ahz + Ely + CZp say; thus hU={[{A-\A) Zz + {B-\B') hy+{C-\C') hp^'^Xdx. By the usual process of integration by parts we get hU={C-\C')hj + \hz + \U-\A' -^hzdx + ^\B-\B' -^^{G-\C')\^ydx. Now assume \ such that ax then, in order that SZJmay vanish, we must have also B-\B'-^{G-\C')=Q. ax Between the last two equations we must eliminate X, and thus we shall obtain a diff'erential equation for determining the required BEUNACCI. 235 relation between x and y. We now proceed to examine whether U is thus rendered a maximum or a minimum. The terms o£ the second order in S F are say =lF{8zY + GSzBy + mzS2)+ll{S^Y+KS7/S2} + ^X (S/j)^ We shall denote the similar terms in BZ by lF'{BzY+G'SzSy + E'SzS^ + ]-I\S^y + K'87/Sjp^lL'{Spy.- Let F-\F' = M, G-\G' = N, H-\H' = 0, I-\r = F, K-\K'=Q, L-\L' = R; then we have to examine the sign of \[m{BzY + 2NBzhij + 2 OBzBp + P(%)'+ 2 ^Sy S^^ + ^ (Sj?)4c/a;. Now assume that this expression can be put in the form I {hijY + mBy Bz + n {BzY + \r [Bp + liBy + UzY dx ; differentiate both sides of the assumed identity, and equate the coefficients of like terms, observing that -j^ can be expressed in terms of By, Bz, and Bjj, since -j- = BZ; thus we shall obtain five equations for determining the five quantities h, Jc, I, m, n, and three of these five equations are differential equations of the first order. Then we assume, as Legendre does, that by giving suit- able values to the arbitrary constants we can make the integrated part I {SyY + mBy Bz + n [BzY vanish. Thus finally if B be always negative between the limits of the integration, we obtain a maximum 236 LEGENDEE, BRUNACCI, JACOBI. value of U, and if R be always positive between the limits of integration, we obtain a minimum value of U. And This is Brunacci's result, and it shews that Legendre's result is wrong. The investigation is of course subject to the excep- tions that have been already indicated in Articles 5 and 6. 207. We now pass to Brunacci's second memoir. This is entitled. Memoir on the criteria which distinguish maxima from minima in double integrals ; it is published in the Memorie delV Istituto Nazionale Italiano, Vol. II. part 2. Bologna, 1810. It extends over pages 121 — 170. Brunacci refers to Legendre's memoir on the criteria for dis- tinguishing maxima from minima in single integrals, and his own correction of one of Legendre's results in his former memoir. He states that so far as he knew, no similar investigations had been made with respect to double integrals. He proposes to consider this point; but before doing so, he gives some investigations with respect to single integrals in order to prepare the way. His memoir is divided into twelve sections. 208. In his first section, Brunacci makes a few remarks on the conditions necessary for the existence of a maximum or mini- mum value of a function. He says that he has proved in his rb Course of higher analysis, that fix) dx is a positive quantity provided that /(a;) is always positive for values of x between x=a and a? = 5, and provided also that the differential coefficients/' (a;), f"{x), ... are always finite between the same values. It is ob- vious however that Brunacci is wrong in saying that it is necessary that the difierential coefficients should be finite; it is sufficient that f{x) be always positive. Brunacci repeats this unnecessary restriction elsewhere in his memoir, but it does not affect his results. 209. In his second section Brunacci investigates the conditions BRUNACCI. 237 which must subsist in order that / yjr dx may have a maximum or a minimum value, where i/r involves x, y, and -^ ; and he shews how to distinguish between a maximum and a minimum. The investigation is the same as Legendre's ; see Art. 198. In his third section Brunacci supposes that y^ involves x, y, -r- and vi 5 and he investio;ates the condition that must subsist dx dot? ° in order that j-\}r dx may have a maximum or a minimum value ; and he distinguishes between the two cases. The investigation is similar to that already given ; and the result coincides with that found by Legendre, and stated in Art. 199. 210. In his fomih section, Brunacci makes some introductory remarks on the subject of double integrals. He states that a double integral \\F{x,y) dxdy taken between definite limits is positive, provided that F{x,y) is positive between the limits of the integrations, and provided also that the partial differential coefficients of F{x,y) with respect to x and y are all finite be- tween those limits. The restriction with respect to the differen- tial coefficients is unnecessary. It is of com'se quite true that in the questions treated by the Calculus of Variations, such restric- tions occur, because certain expansions are effected by Taylor's Theorem ; but Brunacci is wrong in saying that these restrictions occur in the simple case indicated above. 211. In his fifth section, Brunacci takes the integral Uy^dx dy, where -y^ involves x, y, and z, and it is required to determine z as a function of x and y so that the double integral may be a maximum or a minimum. He arrives at the following results; for a maximum or a minimum we must have -^ = 0, and then dz d^^{r^ — x^) . He says that then by determining suitably the first arbitrary function we shall have z = sJ[M+ N {x + ay)+L{x+ ayY] +F{x-ay), where M= \ \, , (1 + am) ,, 2an T 1 (1 + am)^ ' (1 + am)' ' But the surface Brunacci thus obtains will not pass through the curve in question \i F{x — ay) is still left arbitrary. In continuing the discussion of this example he arrives at the result that there is neither a maximum nor a minimum. He says that this ought to be the case, because as -v^ is zero the double integral //-v/r dx dy over assigned limits is also zero. Since he says that -v/r is zero, it would appear that he supposes F{x — ay) = 0, for then his sm-face does pass through the curve in question and i/r is zero. But then it 16 242 LEGENDRE, BEUNACCI. JACOBI. is not obvious what lie means by saying tliat there ought to be neither a maximum nor a minimum, since it is quite possible that a zero value of a function may be a maximum or a minimum value of that function. 215. In his eighth section Brunacci supposes that a/t involves dz dz d^z d^z ^ d^z , , j • xt same manner as before to determine the conditions necessary in order that //-\/r dx dy may have a maximum or minimum value. He arrives at the same results as Delaunay afterwards gave in his memoir; see Art. 147. 216. The remainder of Brunacci's memoir consists of four sections and is devoted to the investigation of the conditions for a dz maximum or minimum of jj-^ dx dy, when -\|r involves x, y, z, -y- , dz and 3- , and also another function F, which is determined by dV , , , ■ ^ dz dz -^-r T dV rni - -7— = 9, where

V= 0, which is linear in Sy, must have no integral By which satisfies the conditions to which by the nature of the problem Sy is subjected. Thus we see that the equation 8V= plays an important part in these investigations, and we soon perceive its connexion with the differential equations which must be integrated in order to obtain the criteria for maxima and minima. Also we easily see that a partial differential coefficient of y with respect to any constant which occurs in y as the solution of V=0, will be a suitable value of Sy for satisfying the differential equation S V= 0. Thus the general expression for Sy as the integral of the equation SV=0 will be a linear function of all the partial differ- ential coefficients of y with respect to the constants which it involves. 224. The equation S F= 0, of which we can thus find the complete integral, can be put in the form of the above equation jACOBi. 249 F= 0, with hy in the place of y. By means of the properties of equations of this kind, we can by repeated integration by parts transform the expression \hVhydx into another, which contains a perfect square under the integral sign ; we thus obtain the trans- formation of the second variation which was always desired. Take for example the int^'ral considered already and let u and u^ have the meanings already assigned. Z V can be put in the form 5,-.^ .. d.A.hy' d\AJy" and 8 F will = when By = u. Now put By = uB'y ; then from the general theorem (Art. 222) we have I B VBy dx= \uB VB'y dx Denote the last integral by I V^ ^y dx ; then the equation I^ = is satisfied when we put B'y = —^, and therefore B'y' = — J — ^-i- . We can now continue the same method by putting y = \^ ^y\ so that by the same general theorem [V^^y'dx=[v^ ''''^ ~/^''' ^'ydx = CB"y'^'y-jC{S"yrdx; and this is the last transformation, in which the arbitrary variation 250 LEGENDRE, BEUNACCI, JACOBI. occurs under tlie integral sign only in the form of a square. And it is easily seen that B=u'A„, ^^ /^^<-^^Y ^ C = — J '— A^. and therefore Moreover A^ — -=-^^ so that C has always the same sign as dy ~j^ has, and this sign must be always positive for a minimum and always negative for a maximum. We must moreover examine whether S"?/' can become infinite within the limits of integration; this we can ascertain by our knowing the functions u and u^, and these we know as soon as the complete integral of the equation F= is given. 225. Although the analysis just indicated requires a good knowledge of the Integral Calculus, yet the criteria thence obtained for determining whether a solution gives in general a maximum or minimum are very simple. I will consider the case in which we have under the integral sign y and its differential coefficients up to the w***, and where the limiting values of x, y, y , y" , ... ^z*""^* are given. Now the 2n arbitrary constants which occur in integrating the differential equation of the (2w)*^ order are to be determined by means of the given limiting values ; but as this involves the solution of equations there will be in general several systems of values for the arbitrary constants, so that several curves may be found which satisfy the same differential equation and the same limiting con- ditions. Let one of these systems be chosen, and let one limiting point be considered as fixed, and then let us pass from this point along the curve to following points. Now take one of these following points as the second limiting point ; then, as stated above, it may happen that through this and the first fixed point a second curve can also be drawn which satisfies the same differential equation as the first curve and has the same limiting values of y' , y", ... 3/'""^'. As soon then as by passing along the curve we arrive at a point for which one of the other curves coincides with JACOBI. 251 it, or as we may say approaclies indefinitely near to it, we have reached the boundary up to which or beyond which the integration must not extend if there is to be a maximum or a minimum ; but if the integration does not extend up to this boundary there will be a maximum or minimum provided that , , [„),2 retains the same sign between the limits. 226. In order to illustrate this by an example I will consider the principle of least action in the elliptic motion of a planet. The integral considered in the principle of least action can never be a maximum as Lagrange believed ; it will not however always be a minimum, but certain conditions must hold with respect to the limits ; these conditions are given by the preceding general rule, and if they are not satisfied the integral will be neither a maximum nor a minimum. Suppose that the planet begins to move from a where a lies be- tween the perihelion and aphelion, and let the other limit be 5, (see fig. 7); let 1A be the major axis, /the sun; then we know that the other focus of the ellipse is obtained by the intersection of two circles described from the centres a and h with the radii 2A — af and ^A — 5/ respectively. The two intersections of the circles give two solutions of the problem which can only coincide when the circles touch, that is when the line ah passes through the other focus. Thus if we draw the chord aa througli the focus /', then by the general rule (Art. 225), the other limit h must fall between a and a if the integral which occurs in the principle of least action is really to be a minimum for the ellipse. If h coincides with cb then the second variation of the integral cannot become negative, but it can become zero, so that the variation of the integral is then of the third order, and so may be either positive or negative. If h falls beyond a then the second variation itself can become negative. If the starting point a is between the aphelion and the peri- helion then the extreme point a is determined by the chord of the ellipse drawn from a through the sun /, (see figure 8) . For if a and d are the limits we can obtain an infinite number of solutions by the revolution of the ellipse round ad. If then in the last case the 252 LEGENDRE, BEUNACCI, JACOBI. second limit falls beyond a there will be a mrte of double curvature between the two given limits for which ^vds is less than it is for the ellipse. 227. I will say a few words on the variation of double inte- grals; the theory of this subject is susceptible of greater elegance than it has obtained even after the labours of Oauss and Poisson, In order to give an example of the way in which it seems to me proper to express the variation of a double integral, I will take the simplest case and consider hjjf{x, y, z,;p, q) dxdy where ^ = _ q =■ — . Let 10 be the variation of z ; then will ^ dx"* ^ dy ^jJA^, y. -,I>, 2) dxdy=^jjdxdy (1*^+ f ^ + f ^' Now the method employed in single integrals consists in this ; the expression under the integral sign is divided into two parts, one £)f which is multiplied by w and the other is the element of an inte- gral. The first must be put equal to zero under the sign of inte- gration if the variation is to vanish ; the second can be integrated and we make the integral vanish. So in like manner I divide the expression under the double integral sign into two parts, one of which is multipled by w and the other is the element of a double integral as follows ; let u = aw and put df df dw df dw _ . du dv du dv dz dp dx dq^ dy dx dy dy dx ' ^ , . dw dw ., Equate the terms m w, -j- , -j- ; thus df . da dv dadv df _ dv df _ dv ^ dz ~ dx dy dy dx 'dp dy' dq dx * hence A=^^-^^-^^- dz dx dp dy dq if this be put equal to zero we obtain the known partial differential equation, which is here deduced in a perfectly symmetrical manner. The function v must satisfy the equation df dv df dv^_^ dp dx dq dy jACOBi. 253 If we put -4 = 0, we have ^jffi^, y. ^,I>: q) dxdy =\\dxdy (-1 ^ _ ^ ^) =\\dvdu, and this taken throughout the given limits must vanish. If z is given at the limits w is zero at the limits and therefore also aw^ that is, u; therefore jjdvdu is zero. If the values of ^ at the limits are entirely arbitrary v must vanish at the limits, or if v = represent the limiting curve the arbitrary functions wliich occur in the solution of -4 = must be so determined that dfdv__^^dv^^^^^ dp dx dq dy '' ' 228. To return to the maximum and minimum ; it is to be regretted that so much confusion prevails in the use of these words. Sometimes an expression is said to be a maximum or minimum when all that is meant is that its variation vanishes, sometimes when it really is neither a maximum nor a minimum. Sometimes an expression is said to be a maximum when all that is meant is that it is not a minimum. Thus Poisson says in his treatise on Mechanics that the shortest line on a closed surface between two given points can be a maximum ; but it is obvious that by inde- finitely small inflexions we can increase the length of any such line however long. In fact the shortest line will only be really a minimum when the general condition laid down is fulfilled (Art. 225) ; that is, when between the two limiting points of the curve two others cannot be found which can be joined by another such curve indefinitely close to the first. In other cases the shortest line is not indeed a maximum ; it is neither a maximum nor a minimum. For surfaces which have at every point opposite cur- vatures I have demonstrated that the shortest line between any two points is really a minimum. [By the sliortest line in the above paragraph is meant the line which is furnished by the- ordinary rules of the Calculus of Varia- tions ; the investigation of it is given in most treatises on the sub- ject, but these treatises do not determine whether the line called the shortest line between two points really is the shortest line between those points. Such a line is also called a geodetic curve.] CHAPTER X. COMMENTATORS ON JACOBI. 229. We now proceed to give an account of the commen- taries and developments which have arisen from Jacobi's memoir. In the sixth volume of Liouville's Journal of Mathematics, dated 1841, there is an article bj Y. A. Lebesgue entitled Memoir on a Formula of Vandermondes and its application to the demon- stration of a Theorem of Jacohis. It extends over pages 17 — 35 of the volume. It begins thus — The principal object of the following pages is in the first place to demonstrate the identical equation ^ d\Ai{tyY ^ _^ d\B^t^^ y^ dd- ~^ dx' ' both the summations are taken from z = to i=n\ {tyY denotes dHy d^t -~ and i''> denotes -^t; y, t, A^, A^^, ... A^ denote any functions of x; Bg, B^,...BnSire functions oi y, A^, A^,... J.„ and their differen- tial coefficients. In the second place we propose to find the law of the functions B^, B^, ... B^. The above words indicate the object of Lebesgue's article. The investigations are rather complicated and difficult to follow ; they ■ depend partly upon the knowledge of the condition of {ntegrdbility of a function. 230. In the same volume of Liouville's Journal there is an article by C. Delaunay entitled Essay on the distinction hetween maxima and minima in questions which depend upon the Calculus of Variations. This essay is in fact a commentary upon Jacobi's memoir ; it extends over pages 209 — 237 of the volume. DELAUNAY. 255 231. Delaunay first proves tlie theorem enunciated by Jacobi, which vfe have given i^ Art. 222 ; Delaunaj's proof is somewhat complicated but perfectly intelligible, and it does not assume a knowledge of the condition of integrahility of a function. It may be observed that the result obtained by Delaunay might be stated more distinctly than he has himself stated it. He really proves the following theorem ; whatever functions of x the symbols u, y, A^, A^, ...An, may denote, it is possible to take 5^, h^, ... 5„ such functions of x that ^ "^"^ dx''' ^ "' dx'" ^ "^ ^'^ dx"^ ^^ ^TJ^ = ^" .i-.r. +yut where the summations denoted by % relate to the letter m and extend from m = 1 to m = n both inclusive. ISFow add A^x^y to both sides of this identity, and suppose d-A ^ then dx"^ dx'" A^u'y+tu ^pj = %+^ ^^m (!)• If now u be taken so that 5„ = the riffht-hand member of this o identity is immediately integrable, and by integrating we have. JY-^u^y + tu '^''Ux^t—ry^ (2). (XJO J iX/Ju Thus Delaunay first establishes the general identity (1) and then deduces (2) which is Jacobi' s theorem enunciated in Art. 222. This is in fact the same order of demonstration as that chosen by Lebesgue. Delaunay's demonstration has been adopted in sub- stance by subsequent writers on the Calculus of Variations ; see the works of Jellett, Price, and Stegmann. It should be observed that in equation (2)- since it has been obtained by an integration an arbitrary constant ought to be 256 COMMENTATOES ON JACOBI. explicitly added to the right-hand side or else supposed to he im plicitly involved on the left-hand side. 232. Delaunay next proves that 8 F can be put in the form which Jacobi gives ; see Art. 224. Delaunay then investigates in full the terms of the second order in the variation for the two cases which Jacobi specially considers, namely jf{^, y^ y) ^^ and jf{x, y, y, y") dx. A mistake occurs in this part of Delaunay' s memoir which should be noticed ; it is on his page 222, and has passed from De- launay into other writers. We will here notice it in the form in which it appears in Mr Jellett's work, since that will probably be most accessible to the reader. On page 95 of Mr Jellett's work he has the following ec[uation !■■ ^ ^ , -r, dh y ^ dx ^ dx dx and he says that any value of Sy which makes S^ = will also make d.B^ T^ dh'y ^ dx^ B — — + ^^— ^^—— vanish ; the true inference ought to have been that any value of Zy which makes 8/3 = will make d.B. d'h'y dh'y '' dx" equal to a constant. This constant will not be zero unless a rela- tion is established between the constants which are involved in the vakie of h'y. That is, in Mr Jellett's notation the four constants (7 , G , G , G^ are not all arbitrary, for such a relation must exist among them as to satisfy his equation {d) and thus reduce them to three arbitrary constants ; and this should be the case since equa- tion {d) is a differential equation oHhe,third order. In fact Delaunay by this mistake omitted that part of Jacobi's memoir which forms the latter part of Art. 221, in which Jacobi DELAUNAY, 257 intimates tliat his results will really involve no more arbitrary con- stants than they ought ; whereas in Delaunay's process there would be too many arbitrary constants. It is possible that the mistake may have been introduced through Jacobi's statement given in Art. 224 that V^ = is satis- Of tied when we put h'y = — ; but Jacobi h^s expressly said a little before that u and u^ are to have the meanings already assigned, and when u and u^ were introduced in Art. 221 it was stated that the constants occurring in them were subjected to certain relations. 233. Delaunay next considers the case in which questions of relative maxima and minima are proposed. Mr Jellett says on page 363 of his work with reference to this part of Delaunay's memoir, " the reasoning does not appear to me to be quite satisfactory, and the conclusion is far less perfect than in the case of absolute maxima and minima." 234. Delaunay examines fom- problems as examples of Jacobi's criteria. 1. The shortest line between two points. 2. The brachis- tochrone. 3. The curve of given length which includes a given area. 4. The curve of given length which has its centre of gra- vity highest or lowest. 235. Lastly Delaunay demonstrates the statements made by Jacobi respecting three differential equations given in Art. 221. It may be observed that Jacobi's memoir involves two points. We have on the one hand Jacobi's own method of exhibiting the criteria for the maxima or minima values of an integral ; this is described by Jacobi in Art. 224, and it is explained by Delaunay in his pages 209 — 234. On the other hand since the method of Jacobi does solve the problem in question, it may be inferred that his method will really supply the solution of the complicated differ- ential equations on which Legendre had made the problem depend ; this is in fact what Jacobi states in Articles 220 and 221, and what Delaunay explains in his pages 234 — 237. It should be remarked that Delaunay here notices the relations which must exist among the constants, according to Jacobi's observation at the end of Art. 17 258 COMMENTATOES ON JACOBI. 221. The second point in Jacobi's memoir will thus be seen to be- long rather to the subject of differential equations than to that of the Calculus of Variations. 236. Delaunay's memoir is interesting and valuable and de- serves especial attention as being the first which gave a demonstra- tion of the whole of Jacobi's method. We have however not thought it necessary to reproduce the investigations because they have been substantially adopted by writers whose works are readily accessible; see Art. 231. 237. In the Journal de VEcole Polytechnique, Cahier28, 1841, there is an article by M. J. Bertrand entitled, Demonstration of a theorem of M. Jacohi; the ai'ticle extends over pages 276 — 283. The theorem in question is that given in Art. 222. Bertrand' s article was published in the same year as those of Lebesgue and Delaunay, but whether it preceded them both, or followed them both, or came between them, does not appear. The proof given by Bertrand depends upon a knowledge of the condition ofintegrability of a function ; the proof is valuable, and as it seems possible to pre- sent it in a clearer form than Bertrand has done, we shall exhibit it here with some modifications. 238. Let «(,, a^, a^, ... a^, denote any functions of a? ; let y be any function of x, and let y, y'\ . . . ?/'"', denote the successive differ- ential coefficients of y with respect to x. Then a differential expres- sion of the following form we shall call a differential expression of JacoM^sform, day d\y" d\y"' d^'a^y (») «o3/+ ^^ + -^-+ -7^^-+ .-. + dx dx^ dx^ '" dvf" ' and we shall denote this function of x, y, and the differential co- efficients oiy, hy cf){y) ; and (v) will denote what the expression becomes when y is changed to v. We shall now prove the following theorem ; let ?; be a quantity such that v(}){y) is an exact differential coefficient, then it is necessary and sufficient that v should satisfy the differential equa- tion (ji {v) = 0. BEETRAND. 259 By saying that v0 {y) is an exact differential coefficient we mean that v^ {y) will result from differentiating with respect to x some function of x, y, and the differential coefficients of y, this fanction remaining imchanged in form whatever may be the value of y in terms of x. We have vj> (y) = v(f> (y) -y(f> {v) + y(f> (y). Now v{v) is an exact differential coefficient, and there- fore v(f) iy) —y(t> {v) is an exact differential coefficient. Since then v(f> [y] — y(f) (v) + y(f> {v) is to be an exact differential coefficient, and v(j>{y) —y(fi{v) is such, y d^ d^^'-^y dy in order that this differential expression may take Jacobi's- form the coefficients must agree with those in the developed form of 4^{y). This requires that we should be able to find ^nj ^ra_i5 ^n_2 7 • • • ^o 5 ^^ ^^ ^0 satlsfy thc following equations; n (n— 1) d^a^ _ 7i{n—l) {n—2) d^a„ . ^vC?a„_j_ It will not be necessary for us to do more with respect to these equations than to observe the following two points. The first, third, fifth, ... of these equations will determine successively a„, «„-i, «,i_2j ..• «o> whatever the coefficients Cg^, C2„_2, Cg^^, ... c^, c^, BERTRAJiTD. 261 may be ; and they assign a single definite value to each of the coefficients «„, «,j_j, ... a^. The second, fourth, sixth, ... of these equations will then give relations involving Cg^.^, Cg^g, ^3^5, ... c^,c^, whicli these coefficients must fulfil in order that it may be possible for the proposed differential expression to take Jacobi's form. Now let the differential expression which is under consideration be denoted by 1/^ {y) , and let us examine the natm*e of the con- ditions which must be satisfied in order that vyjr(y) may be an exact differential coefficient when v is such that -\/r(y) = and only then. In order that ua/t (?/) may be an exact differential coefficient it is necessary and sufficient that dx^-" dx'''-^ ^ dx'""-' dx +^0^-^- This follows from the known condition for the integrahility of a function which will be given hereafter in this work. It may also be deduced from a known theorem in the differential calculus, namely d'y _d\jz ^ d"-^ f dz\ r(r-l) d^ f d^z \ ^ dx'- ~ dx' ^' dx"--^ [^ dx)'^ 1.2 dx'-^' [^ dxV - +^-')^^- Put c^ V for z and use this theorem to transform every term in vyjr [y] ; thus we shall- find that vy\r {y) consists of a series of terms each of which is an exact differential coefficient together with the term Gy, where ^~ dx'-' dx'""-' "^ daf''-^ dx "^ "'"'"• Therefore v^{y) cannot be an exact differential coefficient unless = 0. Or we may obtain the result still more simply thus. Integrate by parts the terms of v^^r {y) as much as possible ; thus we shall find f vi/r (y) dx= 8+ I Cy dx, where S represents a series of terms free from the integral sign. 262 COMMENTATOES ON JACOBI. Hence, as before, v-^^y) cannot be an exact differential coefficient unless 0=0. Now by hypothesis the values of v which make C=0 must be those, and those only, which make '»|r(i?)=0; and therefore the differential equation 0=0 must be identical with the differential equation y^r {v) = 0. Hence comparing the coefficients of the various differential coefficients of v we must have the following relations satisfied, dx ~ ""^ "" "^ ' % 2n (271-1) d'c.^ _ > dc^, _ lT2~ dx' ^^ ^' dx +^2-2-C2«-2, 2n{2n-\) [^n-2) d^c^ _ {2n-\) {2n-2) d%^^ . TTTTS dx' 1.2 dx' "^^ ^ dx It will not be necessary for us to do more with respect to these equations than to observe the following two points. The second, fourth, sixth, ... of these equations will determine successively C2}i_ij <^2n_3j C2re-5 5 ••• <^3j ^li ^^ tcrms 01 Cg^, C3„_2, Cgn,^, ••• Cg, c^', ancl they assign a single definite value to each of the coefficients c^n-i, c^n-z, C2n-^, '" C3 , c, . Thc first, third, fifth, ... of these equations will then give relations which these quantities must satisfy, and by substituting the values of these quantities the re- lations will only involve the coefficients with the even suffixes. It is however certain that these relations will then be identically satisfied; because if they were not it would follow that some necessary conditions must hold among the coefficients with even suffixes in order that v^ {y) may be an exact differential coefficient when V satisfies •\/r (v) = and only then. But this is impossible ; because by the former part of the present article we know that whatever the coefficients with even suffixes may be, if the others are properly determined, "^{y) will take Jacobi's form, and there- fore v^{y) be an exact differential coefficient when v satisfies -^{v) = and only then. BEETEAND. 263 Hence we infer tliat exactly tlie same conditions must hold whetlier we require that vyjrly) should be an exact differential coefficient when ylr^v) =0 and only then, or whether we require that t/t (y) should be capable of being put in Jacobi's form. For we have proved in the preceding article that when the second of these properties subsists the first follows ; and we have proved in the present article that to ensure either the first or the second pro- perty, each coefficient with an odd suffix must have a single definite value in terms of the coefficients with even suffixes which are themselves arbitrary. Thus we have proved, as we proposed, the converse of the theorem proved in the preceding article. 240. We shall now prove Jacobi's theorem given in Art. 222. Let (j) [ty) denote what <^ {y) becomes when ty is put for y ; and let Y denote y^ity) — ty^{y). Then Y is an exact differential coefficient whatever y may be ; this may be shewn in the same manner as that in which it is proved in Art. 238, that v^ {y) — y(f) (v) is an exact differential coefficient. Let Z stand for the integral of Y and let k be an arbitrary constant. Suppose -y- to be a quantity such that -j- {Z— Jc) is an exact differential coefficient. We have = z [Z— li) — \z Ydx ; thus if ;t- {Z—k) is an exact differential coefficient sF is so also. But zY=yz{^{ty)-t<^{y)}, and ^{ty) —t(f){y) is of Jacobi's form with respect to u and its differential coefficients, where u = ty. Hence, by Art. 238, if 2; F is an exact differential coefficient, yz must be one of the values of u u „ dz d fu\ found from <^{u)--j> {y) = 0, say tjz = u, ; therefore ^ = ^ (^ - j • 264 COMMENTATOES ON JACOBI. Thus the multiplier of Z— h which will make the product an exact differential coefficient is a quantity of which the type is -,- f — j . We must now indicate some properties of the expression Z—k. It will be seen on examination that Y does not contain t itself but only the differential coefficients of t; this will also be the case with Z, which is the integral of Y. For suppose the differential expression Y when arranged ac- cording to differential coefficients of t to take the form A §^ A ^ A ^'"^ then if Z contained t at all it could be only by reason of the term dZ dx dZ Ajt entering into Z; and then -^ or Y would contain the term dA dA t —7-^ . And —j-^ is a function of ?/ which is at present quite dA arbitrary, so that -j^ cannot be zero. Thus as Y does not con-" tain t but only its differential coefficients, it follows that Z does not contain t but only its differential coefficients. We may shew that Z does not contain t in another way. If we integTate each pair of terms in Y in the manner given in Art. 238, we find that ^consists of pairs of terms of which the type is jcZ^ d'-^aXtyT _ d^ dna^^ \dx^ ddd * dx^ dx''~' and on effecting the differentiations we see that t does not occur in this expression but only differential coefficients of t. dt If then we put r for -j- the expression Z-k will be a differ- ential expression of the order 2w - 2 with respect to t and its differ- ential coefficients. And the solutions of Z~ k = can only be such quantities as render Z constant and therefore Y zero ; that is, the values of t must be those of which the type is 7^ ( — ) • BERTRAND. 265 Tlius Z— Z; is a differential expression sucli tliat any multiplier of it which renders the product an exact differential coefficient must be a solution of Z—'k = 0. Hence, by Art. 239, this differential expression must be capable of being put in Jacobi's form ; that is, omitting the arbitrary constant, the integral of y(f>{iy) — iy4>{y) is of the form ' ^ dx "^ dx' "^ — "^ dx''-^ ' If now we suppose y such that ^[y)'= 0, we have the integral of y^ [ty) assuming the above form. This is the theorem enunciated in Art. 222. 241. A remark may be added to obviate a possible miscon- ception of part of the preceding article. The equation Y= is of the order 1n — \ in r and its differential coefficients ; thus the general solution of it will involve 2n — 1 arbitrary constants. This general solution would make Z constant since it makes -^— = ; ° dx therefore in order that Z—k may be zero a relation must hold among the 2n — 1 arbitrary constants. Thus in effect we have only 2n — 2 arbitrary constants in the solution of Z— h = 0, as of course should be the case. Particular solutions of Z— Jc = will then be obtained by giving particular values to any or all of these arbitrary constants. 242. More than ten years elapsed before another commentator upon Jacobi's memoir appeared. We have next to consider a memoir published by Professor G. Mainardi in the third volume of Tortolini's Annali di Scienze Mathematiclie e FisicJie, 1852. This memoir is entitled Researches on the Calculus of Variations; it occupies pages 149 — 192 of the volume, and there is an appendix which occupies pages 379 — 383. 243. Mainardi begins by referring to what had been done by Poisson, Ostrogradsky, Cauchy, Sarrus, Jacobi, Bertrand, Lebesgue and Delaunay ; he intimates that the propositions of Jacobi require yet to be more completely developed, and he says that with respect to the criteria which distinguish a maximum from a minimum in 266 COMMENTATORS ON JACOBI. the case of multiple integrals he believes nothing had been added to the remarks of Legendre and Lagrange. 244. Mainardi's memoir is divided into five sections. The first section occupies pages 149 — 153 ; this section contains some illustrations of the method which was used hj John and James Bernouilli in solving isoperimetrical problems. Poisson in his memoir had referred to this old method, see Arts. 88 and 97 ; and Mainardi intimates that he will hereafter publish his researches on the comparison of the old and modern methods. He confines him- self in this section to shewing how the old method could be made to give the terms relative to the limits in the case of a single inte- gral, and how it could be made to give the variation of a double integral. 245. The second section occupies pages 154 — 171. Mainardi says that in this section he proposes a new method for distinguish- ing between the maxima and minima values of integrals. Speaking generally this method may be described as Legendre's improved by some additions borrowed from Jacobi. Mainardi considers suc- cessively six cases. (1) A single integral involving x, 7/, and -^ . (2) A single integral involving x, y, -f-, and ,-■; . (3) A single integral involving x, y, -^, -j4 5 ^^^ A • (^) ^ double integral involving x, y^ z, -7- , and -^ . (5) A double integral irtvolving dz dz d'^z d^z , d'^z v 1 , , ^i ="' ^' ^' S' ^' ^' SS^' '""^ ^' ■'' "''o ^"'^^^'^ '^P"" *''' particular case in which the double integral involves only a?, y^ z, -^ , -^ , and -j — 7- . (6) A single integral involving x, y, z, ~ , dx dy dxdy ^ ° ° ' -^ ' rfa; ' dz an(J -_ . Of these cases (1 ) and (4) may be considered to be com- dx pletely investigated, (2) and (3) nearly completely, and the others only imperfectly. We shall presently give a more detailed account of some of these cases. MAINARDI. 267 24'6. In his third section Mainardi gives an investigation of Jacobi's theorem enunciated in Art. 222, using, as he sajs, Bertrand for his guide. This investigation extends over pages 172 — 179, and then Mainardi indicates brief! j the application of the theorem to the Calculus of Variations. Mainardi's proof does not seem so good as Bertrand's; the principal difference consists in replacing the indirect reasoning of Art. 239 by direct reasoning. But a student who had not read Bertrand's proof would find one point of Mainardi's unsatisfactory. For on comparing, as we have done on page 262, the coefficients of the various differential coefficients of v, Mainardi only writes down what we have called the second, fomlh, sixth, equations; and he says briefly that these include the others; see his page 177 at the top. This amounts to omitting one of the most difficult points in the investigation. 247. In his fourth section Mainardi applies Jacobi's method to a double integral ; this section extends over pages 183 — 185. There is no difficulty in his first case where differential coefficients of the first order only occur; but in his second case where differential coefficients of the second order occur Mainardi himself intimates at the end of the section that he has accomplished very little. 248. The fifth section extends over pages 185 — 192. In this section Mainardi says that he will collect some applications of the Calculus of Variations which afford ground for some remarks ; accordingly he discusses four examples. (1) He gives a theorem on geodetic curves ; ihis amounts to finding the first integral of the equation which determines such a curve for a large class of surfaces. (2) He speaks of Gauss's theory of capillary attraction as affording one of the finest modern applications of the Calculus of Variations ; but he thinks that the investigation given by Gauss admits of great simplification. Accordingly Mainardi gives an investigation of the variation of the fimction which Gauss con- sidered ; see Art. 71. Mainardi's investigation is far shorter than that of Gauss, but it would not be very easy to follow unless the student had previously read Poisson's memoir or some equivalent method. (3) Mainardi forms the equation furnished by the Cal- culus of Variations for the form of a flexible sm-face which is in equilibrium under the action of gravity ; this problem will be 268 COMMENTATOES ON JACOBI. found in Mr Jellett's Calculus of Variations, page 323. (4) Mai- nardi says that Steiner found by a geometrical method an elegant property of tlie polygon of given perimeter which can be drawn on a given surface so as to have a maximum area. Mainardi infers from the Calculus of Variations that when such a polygon is to be inscribed in a given polygon, the two arcs of the required polygon which meet on a side of the given polygon will there make equal angles with that side. Mainardi gives no reference; a memoir by Steiner will however be found in the sixth volume of Liouville's Journal of Mathematics. Steiner's enunciation of his theorem occurs on page 168, and the enunciation is more explicit than Mainardi's, namely, the two arcs which form a part of the inscribed figure, and meet on the same side of the given figure, either cut it in one point at equal angles or else touch it in two points. 249. We have thus given an outline of the whole memoir, and we shall now return to the second section of it and examine more particularly the method proposed by Mainardi for distinguish- ing between maxima and minima values. The second section con- stitutes in fact the most important part of the memoir, and although it will be seen that the investigations are incomplete, they are not without interest and value. The appendix to the memoir is de- voted to the elucidation of part of the second section, and we shall presently have occasion to refer to it. We may remark that the whole memoir is difficult, and that it is disfigured by extreme inaccuracy of printing. 250. We will first give Mainardi's method for distinguishing a maximum from a minimum in the case of a single integral in- volving X, y, and -J'- . Let \F {x, y, y') dx denote the integral which is to be a maximum or minimum. Change y into y + icc, where i is sup- posed to be an indefinitely small constant quantity and o) an arbi- trary function of x. Then expand the new value of F{x, y, y') in a series proceeding according to ascending powers of /; thus MAINARDI. 269 the new value of the integral, that is \F {x, y + iw^ y +%Qi) dx, is equal to j^ (^j y, y') dx + I^'i+I^-+ ... where the terms not expressed involve powers of i higher than the second; and r [fdF dF \ ^ The expression IJ constitutes the variation to the first order of the proposed integral ; this must vanish, and thus by the usual method we arrive at the equation dF__ddF_ ,^, dy dxdy From this equation we must suppose y to be found in terms of a?, and when this value of y is used, let ^=.A -^lI_^b ^=(7 dy"^ ' dy dy' ' dy'^ We have then to examine the sign of \Aa>'' + 2BoyQy'+Cco'^)dx. /( Now assume that we have identically Aai" + 2Bco(o' + C(o'^ = (ilfo)')' + ao)' + 2l>oia>' + cco'^ ; then we must have a + M' = A, b + M^B, c= C (2). We have thus three equations involving the four unknown quantities a, b, c, M, so that we are at liberty to make one more supposition respecting them ; it is found convenient to introduce another quantity and to make two more suppositions. Let 6 denote this new quantity, and suppose ae + bd' = 0, and 56' + c^' = (3); 270 COMMENTATOES ON JACOBI. thus (2) and (3) supply ifive equations for determining five quan- tities. From the first two equations of (2) combined with (3) we obtain Be+ C6' = Md, A0+B6'= {Mey (4) ; hence A9 + Be'-{Be+C6')' = Q (5). From this difierential equation 6 must be determined, and then from the first of equations (4) we have M=\{B9+Ce'); then from equations (2) we must obtain a and h ; also it appears from (3) that ac = If, so that , , f h V aaf + Ihcao) + cw'^ = c ( w' -| co ) . Also c= G; thus finally \A' + Cio"-) dx^^Moy'+jC (co' + ^ (oXdx, where M has the value just assigned. Hence if G retains constantly the same sign between the limits of the integration, and Mof either vanishes at the limits or gives rise to a result of the same sign as G, we have in general a maximum or a minimum according as the sign of G is negative or positive. It may be observed that -^ = — yf- = — -^ • 251. The above article contains all that is peculiar to Mai- nardi, for the differential equation (5) is solved, with the assistance of Jacobi, in the following manner; see Art. 221. Let the value of y found from (1) be denoted hj f{x, 7^^, 7J, where 7^ and y^ are arbitrary constants ; then we shall have where /S^ and ^^ are new arbitrary constants ; this we shall now prove. The expression 7^ is the coefficient of / in the expansion of MAINARDI. 271 the varied value of the proposed integral \F{x, y, y') dx, and I.^ is the coefEcient of - in the same expansion. We may also say that /g is the coefficient of i in the expansion of the varied value of I^ ; that is, if in I^ we change y into y + ico, and expand in a series proceeding according to ascending powers of i, the term involving * will be found to be I^i. Now if the limiting values of y are fixed, /j will vanish whatever may be the values ascribed to the constants 7j and 7^, so that /, will also vanish when 7^ is changed into 7j + ^7^ , and which makes the coefficient of to under the integral sign in the last form of /^ vanish, must be Thus the value of is to be found in the manner already stated. 252. The solution of the differential equation (5) of Art. 250 does in fact constitute one of the most important parts of Jacobi's theory. We have here had occasion to use it in only a simple case, namely that in which equation (5) of Art. 250 is of the second order; the method however is perfectly general whatever be the order of the equation analogous to (5) , and we shall have to apply it again. The general process is as follows. With the usual nota- tion the terms of the first order in the variation of an integral will take the form/ 1^?/ c?a;, excluding the integrated terms. The terms of the second order, with the same exclusion, will take the form - jSVB^dx, where Now suppose that the solution of the equation F= is where 7^, 73, ... are arbitrary constants. If this value of 1/ be sub- stituted in V the result will be identically zero, so that we may differentiate Fwith respect to any of the arbitrary constants which df occur in /, and the result will still be zero. Let ~- = u, then by differentiating V with respect to 7^ we obtain dV dV , dV „ -J- U + -y-? '^ + T-n U +... = 0. ay ay ay This shews that 8 F= is satisfied by hy = u; and therefore it will be satisfied by Sy = ^u, where 13 is an arbitrary constant. Hence the general solution of S F= will be MAINARDI. 273 253, Next let the proposed integral, which is to be a maxi- . , dy •. d^y mum or ammimum, involve x, y, -¥- and -7^ . Let jF (a?, y, y', y") dx denote this integral, change as before y into y + ica, and expand the new value of F{x^ y, y\ y") in a series proceeding according to ascending powers of i ; then the new value of the integral may he denoted bj jFix, y, y\ y") dx+IJ. + I^ ^ + , , ^ ndF dF , dF ,\ , ((d'F d'F , d'F , d'F , ^ d^F „ ^ d'F , ,\ , + 2 ^ — 7-7, (wci) + 2 , , , „ Q> 0) aa;. «3^ dy dy dy } Then as usual 7j must vanish ; this leads to the equation dF_d_dF ^dF^_^ ,.v dy dx dy dx^ dy"~ From this equation we must suppose y to be found in terms of X, and when this value of y is used let d^F_ d'F _ d^_ ^ df~ ' dydy''^ ' dy"~^' dy'^ ' dy dy" ' dy' dy" We have then to examine the sign of [{Aco' + 2Bco' + Co)'^ + Goi"^ + 2H(oo)" + 2Kaj'(o") dx Hence if Q retains constantly the same sign hetween the limits of the integration, and the integrated part either vanishes or gives rise to a result of the same sign as G, we have in general a maxi- mum or a minimum according as the sign of G is negative or positive. 254. It remains to shew how to determine the auxiliary quan- tities a, h, c, Ji, k, 31, N, P, 6 which are introduced in the preceding article; for if they are not determined we shall not he able to ascertain whether they remain finite or not between the limits of the integration. The value of 6 is determined as before ; see Arts. 251 and 252. If we represent the solution of (1) by 1/ =f{^j Jiy %, %, %) where 7^, y^, y^, y^ are arbitrary constants, we shall have dy, dy^ dy^ dy^ where /S^, /Sj, /Sg, ^^ are new arbitrary constants. It is with respect to the methods which he proposes for determining the remaining auxiliary quantities that Mainardi's investigations are the least satisfactory. He proposes in fact three methods for this purpose. (1) He intimates obscurely that h and k may be determined thus ; let ^ be a quantity found like 6 from the differential equation 18—2 276 COMMENTATORS ON JACOB!. (6) of the preceding article, so that cj) is of the same form as but has other arbitrary constants instead of yS^, ySg, ^^ and yS^; then h and ^ will be determined by the equations O6"+ke' + hd = 0, G(f>" + kf + h(l) = (7). Mainardi seems to intimate that if h and k be thus determined and then the remaining auxiliary quantities deduced from such of the equations (2), (3), (4) as may be convenient, the remainder of these equations will be satisfied. See his page 157 at the bottom. (2) Mainardi however seems to allow that the statements just made require to be proved ; and accordingly he proceeds to verify them. With respect to this verification we may observe that it is really a long process, and in consequence of it Mainardi's method loses the apparent simplicity which constitutes its chief recom- mendation. Moreover in this verification, on the sixth line of his page 168 the right-hand member of his equation should be a con- stant and not zero as he gives it ; this in fact is the same mistake as we have already indicated in Art. 232. Thus k and k cannot be found as Mainardi intimates from equations (7) where the four constants in <^ and the four constants in are all arbitrary ; there must be a relation between these constants. (3) Mainardi returns to the point in the appendix and offers another reason for the statement that h and k are to be found from equations (7). He says the equations (2) of Art. 253 really ex- press the conditions that must hold in order that {A-a)co' + 2 {B-h) coco' J^ {0 - c) co'^ -^ {G - g) co"^ + 2 {H- h) (0(o" + 2 {K- k) an arbitrary function of X and y. Then expand the new value oi F{x, y, z, z\ sj in a series proceeding according to ascending powers of i ; thus the new value of the double integral is jj^(a?5 2/, z, z\z) dxdy^I^i^ ^2 1" + ' where /, =//(g'« + ^o,' + ^^«^) ^^c?3,, 278 COMMENTATORS ON JACOBI. The expression I^ must vanish, and thus in the usual way we arrive at the equation dF _i_dF_d_dF_Q Cll dz dx dz dy dz^ From this equation we must suppose z to be found in terms of X and a/, and when this value of z is used let dz"^ ' dzdz' ' dzdz^ ' dz'^ ~ ^' dzdz, ~ ' dzj' ~ We have then to examine the sign of \A(o'' -f 25coa>' + 2 Ccow, + Gco'^ + 2H(o'o>, + Gco'^ + 2Ha)'oi, + Kwf ; then we must have a + M'-^N, = A, h+M=B, c + N=G (2). We have thus three equations involving the^ue unknown quan- tities a, 5, c, M, N, so that we are at liberty to make two more suppositions respecting them ; it is found convenient to introduce another quantity and to make three more suppositions. Let 6 be this new quantity and suppose that He' + Kd, + ce = i (3). MAINAEDI. 279 From (2) and (3) we olbtain Gd' + Hd^ + Be^Md ^ He' + Kd^+Cd = N0 I (4); B6' + Ce, + A9= {MOy + {NO), I hence Be'+c0,+A9-{Gd'+ird,+Bey-{Re'+Ke^ + ce)=o (5). From this partial differential equation 6 must be found ; then from the first and second of equations (4) we obtain M and N, namely, M='^{G9'+Ee, + Be), N=^{iie' + Ke^ + ce)', and from equations (3) we obtain a, h, and c. Now we have aoy" + 2hQ}Qi' + 2ccoco^ + Gcd'^ + 2Ra)'(o^ + Kwf H\r Oc-Hb V 03 + (K--^)U + aj\ ' ' KG-H 9 {Gc-my \ . f h' {Gc-my r G- G{KG-H') and the coefficient of w^ in the last line vanishes, as we shall find by eliminating -x and -~ from equations (3). Hence, finally, we obtain [[{Aoi'' + 2B(ow + 2 C&JG), + Geo"" + 2Zra)'a>^ + Ziy/) dxdy The expression in the first of these two lines really involves only a single integral ; the expression in the second line is a double 280 COMMENTATORS ON JAGOBI. integral. In order to ensure that this double integral shall retain the same sign whatever w may be we must have K—-jy and O of the same sign; that is, OK— IP must be positive. Then if GK~E^ is constantly positive throughout the limits of the inte- gration we shall in general have a maximum if G be constantly negative, and a minimum if G be constantly positive. These results agree with those obtained in Art. 213. 257. A value of 6 which will satisfy equation (5) of the pre- ceding article may be obtained in the manner explained in Art. 251. Suppose, for example, a solution of (1) obtained which in- volves two arbitrary constants, and denote it by where 7 and % are the two arbitrary constants ; then the partial differential equation (5) will be satisfied by where jS^ and ^^ are arbitrary constants. 258. Thus it will be seen that the investigation given by Mainardi of the question discussed in the preceding two articles may be considered complete, because the values of the auxiliary quantities introduced can be really found. But the investigation is not preferable to another which Mainardi gives and which exactly follows the method given by Jacobi for a single integral. With this other investigation we will close our account of Mainardi' s memoir. We will suppose, as is usual in discussing Jacobi's method, that the limiting values of z, -7- and -r- are given so that the quantities co, «', and eo, vanish at the limits. With this sup- position it will be found that the expression in Art. 256 of which the sign is to be examined may be written thus, 1 1 l^ft) -1- B(o' + C(o^ - {Boo + Gq)' + Ho))' -{C(o + H(o' + Km) , \ wdxdy. MAINARDI. 281 We may prove this by integrating jj{Bco + Gm + Hm) ' adxdy and I j ( Ceo + Hco' + Kco)^ (odxdy eacli once by parts, and then we shall obtain the same expression as we used in Art. 256. Or we may modify the form of I^ and then deduce that of I^ in the manner explained in Art. 251. Let a stand for A—B'—G/, then the expression of which the sign is to be examined may be written {{{aw -{Gai' + Ha>) ' - [Hco' + K(o) \ (odxdy. Let 6 be such a quantity that 0,6 - {Ge'+Eey-{He'+Kex = o, and assume a> = u6. The above double integral becomes IJLud - ( Gu'O + Gud' + Hufi + Eudy - {Hud + HuO' + Kufi + Kue)\ uOdxdy ; and this on reduction will be found equal to - {{{[{ Gu + Hu) 6''-]' + [{Hu' 4- Ku) eF]\ udxdy. Integrate by parts and omit the terms which vanish at the limits ; thus this double integral becomes /J|( Gv! + Eu) d'u' + {Hu' + Ku) e'uX dxdy, {{[ Gu" + "iHu'u, + Kuf\ mxdy. Hence, finally, we have in general a maximum if GK—H^ is positive and G negative throughout the limits of the integrations, and a minimum if GK— H^ is positive and G positive. The quantity B may be determined in the manner explained in Art. 257. that is, 282 COMMENTATORS ON JACOBI. 259. In the volume of Tortolini's Mathematical Journal which contains Mainardi's memoir there is a short article on our suhject hy Professor F. Brioschi. It is entitled, On a theorem ofJacohis relative to the criteria for distinguishing the maxima from the mi- nima values of integrals. The article occupies pages 322 — 326 of the volume. Brioschi refers to Mainardi's method for distinguishing maxima from minima values, and he says this method is complicated, hy the admission of Mainardi himself. Brioschi then says he will "briefly indicate criteria for solving the problem proposed. Thus the title of the article does not give a correct idea of its con- tents ; for there seems to he no reference to Jacohi's theorem, but instead of that a new method is proposed. 260. Brioschi does not demonstrate the theorems he enun- ciates ; the theorems themselves are enunciated in the language of determinants. The following example will give some idea of the object of the article. Consider the expression Aoi" + B(o^ + Oo)"' + 2^0)0)' + 2Fa)a>" + 2 G(o'(o" ; this expression can be put in the form C (co + ^ J + a (O) + ^(oY + ry(o\ where a, /3, 7 are certain functions of ^, B, ... G; we have in fact indicated the values of a, /3, 7 in Art. 256. The use of such a transformation is that we can thus see what conditions must hold in order that the original expression may be incapable of changing its sign; if a, 7 and C are all of the same sign, or if a and 7 are zero, the proposed expression cannot differ in sign from G. Now the theory of determinants furnishes general forms for such coefficients as we have denoted by a and 7 whatever be the number of the quantities w, (x,y, z, z', z,, ...). Then in the first place we observe that by properly choosing the coefficients A^, A^, B^, G^, A^, ... we can obtain the following identity, zF{x,y,z, z, z^, ,..)-z^{x,y, z,z, z^, ...) =z'¥{x, y,z, z^, ...), where '^(a?, y, z, s^, ...) is a linear function of the differential co- efficients of z of odd orders. In the second place we prove that z^{x,y, z,z, z^, ...) possesses the property which by supposition zF(x, y, z, z', z^, ...) possesses. EISENLOHE. 291 For change z into wt and subtract all that involves f and then divide hj t ; we thus obtain terms which may be arranged in pairs, and bj pairing them suitably we shall obtain expressions which are integrable either with respect to x or y. For example a part of the result is "^ Tx {^^ ^^'^'+^^ ^''^^\ ^""^{^^ ^""^^'"^ ^' ^'''^} and we arrange these terms in the following pairs, the first of these four expressions is exactly integrable with respect to X and the second with respect to i/; the third expression is equal to ^wB.M^-^wtBy, and the fourth to -rz- wB. (wt)' — ^wtB.w,'. ay ^ ^ ' dx ^ ' and thus every term is susceptible of exact integration either with respect to x or y. The general process which is exemplified in the first and second of these four expressions presents no difiiculty ; that which is exemplified in the third and fourth of these expressions will be now given. Let us denote one of the terms in z^{x, y, z, z\ 0^, ...) by <^™ /.. d'^z dx^'dy'X dx^dy")' 19—2 292 COMMENTATORS ON JACOBI. where r + s = /> + cr = w; then, as we suppose r not equal to ^, there will also he the term ^dx^dy^K dx^ d'lfi Now change z into wt and subtract all that involves «^ and divide by t. We thus obtain J- (^ d'^wt \ , d^ (^ d/'w \ ^'^^^ V d^^-r dx^dr \^ dx^df] ' f?™ / ^ d^wt \ ^ d^ /^ drw\ ^""^ '"d^A^d^^)-'^' l^fd^^ K^d^fdf) ' Now by repeated integration by parts we have where 8 represents a series of terms free from the integral sign. Then if we integrate both members of the last equation with respect to y, we shall find that the only term on the right-hand side that remains under the double integral sign is (-1)-//, d'^w r^ d'^wt , , K ^ 7— ay ax. dxl' dy^ dx^ dy° And this term is the only term that will remain under the double integral sign when we integrate wt , „ , , K dx" dy" \ dx^ dy' Hence the first pair of terms written above is such that it consists of parts which are susceptible of exact integration either with re- spect to X or y. And the same holds with respect to the second pair of terms. Thus z^{x^ y, z, z, z^, ...) does possess the property in question. In the third place it will follow that z'^{x, y, z', z^, ...) must also possess the property in question or else vanish identically ; and from this it will follow that '^ {x, y, z, z^, ...) does vanish identically. If for example "^ {x, y, z, z^, ...) does not involve differential EISENLOHR. 293 coefficients of a higlier order than the third, we should have for z"^ {x, y, z, z^, ...) an expression of the form z jjf/ + M,z^ + N,z"' + N,z;' + N,z,: + N,z,,}^ . Hence the following expression must be susceptible of inte- gration with respect to x or a/, or else vanish identicallj, w iM, [wty + M, iwt), + N, {wty + N, [wt)'/ + N, {wt)\^ + N, {wt)\ - wt my + M^w^ + iv; w" + N^ w'/ + N^ w' „ + iV, w\ . By reducing the terms of the first line by integration by parts with respect to x or y, we arrive at an unintegrated expression of the form Gt where C does not contain f; tliis must vanish since it cannot be an exact integral with respect to x or y. And as G must vanish whatever w may be, we shall find in succession that N^, JSf^, N^, N^, ifg, ilfj must all vanish. Thus Eisenlohr's theorem is established. The theorem is applied to the purposes of the Calculus of Vari- ations in a manner similar to the application of the theorem in Eisenlohr's sixth section. 270. The next work we have to consider is by Spitzer, en- titled On the criteria for maxima and minima in problems of the Calculus of Variations. This work consists of two memoirs which were communicated to the Academy of Sciences at Vienna ; the memoirs were published in 1854 in the Sitzungsherichte of the Academy. The first memoir extends over pages 1014 — 1071 of the 12th volume, and the second over pages 41 — 120 of the 14th volume of the Sitzungsherichte. Spitzer refers in the beginning of his first memoir to the memoirs of Jacobi and Delaunay, which Tve have already noticed ; and then he says, that he has sought to deduce Jacobi's criteria in another manner, and believes that this new way may deserve some consideration. 271. The two memoirs consist altogether of thirty sections, of which thirteen are contained in the first memoir, and the remainder in the second. The first section gives the ordinary 294 COMMENTATOES ON JACOBI. » investigation of the'terms of tlie first order in the variation of an integral which involves a?, «/, and the differential coefficients of y with respect to x. The second section gives an investigation of the terms of the second order in the variation of the integral. The third section shews how Legendre transformed the terms of the second order so that the existence of a maximum or minimum might be recognized ; Spitzer writes the equations at full for the case in which the integral involves only the first differential coefficient of y^ for the case in which it involves both the first and second differential coefficients of ?/, and for the case in which it involves the first second and third differential coefficients of y. In his fourth section Spitzer makes some remarks on the equa- tions given in his third section ; he shews that the equations take the complicated form that has been indicated in Arts. 220 and 221, and he says that Jacobi had succeeded in integrating these equa- tions by a refined and difficult analysis, and that he himself had solved the equations in a much simpler manner. The fifth sec- tion contains that part of Jacobi's theory which we have given in Art. 252. The sixth section indicates briefly the general way in which Spitzer proposes to solve the problem under discus- sion. The seventh section contains a complete investigation of the general criteria for the maximum or minimum of an integral Vdx^ where V involves a?, 3/, and y , The eighth section con- /. dW . tains a discussion of that particular case in which -y-pa is zero. The ninth section contains a complete investigation of the ge- neral criteria for the maximum or minimum of an integral I Vdx J Xq where V involves x, y, y', and y". The tenth section contains a d^V discussion of that particular case m which -,-775 is zero. The eleventh section contains a complete investigation of the general criteria for the maximum or minimum of an integral I Vdx where V involves cc, y, y', y'\ and y"\ The twelfth and thirteenth sections contain a discussion of that particular case in which -j-7772 is zero. The fourteenth, fifteenth, and sixteenth sections SPITZEB. 295 contain some additional investigations respecting the particular cases which are discussed in the eighth, tenth, twelfth, and thir- teenth sections. The seventeenth section gives the ordinary in- vestigation of the terms of the first order in the variation of an integral which involves ic, y^ z, and the differential coefficients of y and z with respect to x. The eighteenth section gives an inves- tigation of the terms of the second order in the variation of the integral. The nineteenth section shews how the terms of the second order are to be transformed so that the existence of a maximum or a minimum may be recognised ; the necessary equa- tions are given at full for the case in which the differential co- efficients which occur do not rise above the first order, and for that in which they do not rise above the second order. The twentieth section generalises the theorem given in the fifth section. The twenty-first and twenty-second sections contain a complete inves- tigation of the general criteria for the maximum or minimum of an integral I Vdx where V involves x, y, z, y and z. The re- J Xq maining sections contain discussions of the particular cases which occur when V assumes particular forms. 272. Speaking generally we may describe Spilzer's work in the terms we used with reference to Mainardi's, namely, as Le- gendre's method improved by additions borrowed from Jacobi; see Art. 245. Spitzer was acquainted with Mainardi's memoir, for he refers to it on page 62 of the 14th volume of the 8itzungs- henchte. The investigations of Spitzer however are much more complete than those of Mainardi ; Spitzer does not shrink from the labour of working out the solutions of his equations com- pletely. Spitzer was the first who developed completely the second variation of an integral involving x, y, y', y", and y'"; the preceding writers had confined themselves to the case in which the integral involved only x, y, y' and y". Spitzer' s investigation of this problem is extremely complex, and occupies twenty large octavo pages ; besides seven more pages which relate to certain special cases. In fact it seems improbable that any student would verify the long calculations contained in Spitzer's twelfth and thirteenth sections, and in his sections comprised between the twenty-first and thirtieth inclusive. 296 COMMENTATORS ON JACOBI. It should be olbserved tliat the memoir is well and correctly printed; some mistakes at the ends of sections 8, 10, and 13 are corrected 'by the author himself in "a note to section 14. A mis- take occurs in the second line of page 1032 of the 12th volume of the Sitzungsherichte, for the sign of the right-hand side of the equation must be changed ; this mistake leads to two more on the same page, and it appears again on page 44 of the 14th volume. 273. We will now give some specimens of the investigations and conclusions of Spitzer. In Arts. 250 and 251 we have shewn how in general we may distinguish between the maximum and minimum of I Vdx, where V involves x, y. and y. Now suppose for a particular case that ^-i2 = 0, then, excluding the integrated terms, the value of I^ on page 271 will take the form iydy"^ dx dy dy' ) Thus for a maximum or minimum it is necessary that dy^ dx dy dy should be respectively constantly negative or constantly positive throughout the limits of the integration. d^V Since -r-fi = 0, it is obvious that Fmust be of the form thus the differential equation from which y is to be found, namely, dV_d^dV^ dy dx dy' ' becomes ^+2,' J_g-^y = 0, that is ^-^ = 0. dy dx SPITZER. 297 This gives y as a function of x witto'dt any arbitrary constant. Thus, adopting geometrical language, the limiting points between which the required curve is to be drawn cannot be taken arbitrarily ; they must lie on the curve determined hj —— ~- = 0, or else the problem will be impossible. In the next place let us suppose that besides , ,3 = 0, we have dW d d^V ... ^ also -^-5 5 — ? — 7-, = : Spitzer in his fourteenth section deter- dy ax dy ay mines what the form of V must then be, in the following manner. d'^V Since -r-^ = 0, we must have Fof the form^ (^j2/) +y'fi {^^y)' Now suppose f^ {x, y) and f^ {x, y) expanded in series proceeding according to ascending powers of y, and let fA^,y)=^B, + B^y + B,f-^B^f+.., Thus ^=1.2^, + 2.3^32/ + 3.4^,y+... + y' {1. 2B^ + 2.ZB,y + Z,4.B,f +...)', ^ = B, + 2B,y + ZB,f + ^B,f +...', And since + y (1.2^, + 2. 3^33/ + 3. 45,/+...). d'V d d'V dy'^ dx dy dy' ' we must have 2A, = B^, 3^3 = 5;, 4^^ = 53', ... Thus F=A + A3/ + §^y + §^/ + ^/+... + y'{Bo + B,y + B,f + B,f + B,y^ +...), or F=A + Ay + 5oy+(§2^« + |V + fy + ...)'. 298 COMMENTATORS ON JACOBI. This gives for V the following form, {x) +y,{x) +y',{x,ij)]'; or we may express our result without any loss of generality thus, , With this value of V we have jVdx = ir{x,y) +jyx[^) dx ; thus in h I Vdx the unintegrated part is \x{x) ^ydx, and this will not vanish unless % [x) vanishes. Then / Vdx is exactly integrahle, -and its maximum or minimum can l)e sought hj ordinary methods. 274. Besides Spitzer's method, we may use another for finding the form of V in order that we may have dW ^ ' , d'V d d'V ^ ^-^ = 0, and also -^ - ^ ^-^-= 0. The latter result is the condition of integrahility of the function -:=- ; so that we must have ^— an exact differential coefficient of dy _ dy some function of x and y. Thus We do not introduce y' into the function f{x,y), because if dV we did -T— would contain y"', and this is impossible, because since d'^V ^ -—2=0, we know that V is of the form f{x-,y) -^ y f^ix^y), '^ dV and so -5- does not involve y". dy ^ Thus ^=i^+V^ dy dx dy '' therefore F= l-f^ dy + y j-^ dy. SPITZER. 299 Now let Fioe, y) be such a function of x and y, that dy ~]dy ^' ■where %i (a?) is an arbitrary function of x. where pj;2 (a?) is another arbitrary function of x. Therefore ^ = ^ + 2/' "^ + 3/%/ (^) + %. («^) = |^(«^; y) +jx^ {^) dxj + yxl (^)- And this agrees with Spitzer's form of V. 275. We will in the next place shew the manner in which Spitzer investigates the criteria for the maximum or minimum of / Vdx, where V involves x, y, y' and y". We have first to find in the ordinary manner the terms of the first order in 8 I Vdx and to make them vanish. Then to distinguish between a maximum and a minimum, we must investigate the sign of j\dy dy dy dy dy , o d'^V „ ^ d'^V , „\ , + 2 J J „ WW + 2 , ,j „ w to ] dx, dydy dy dy J ' where lo is put for hy. ^ ^^ dV dV , dV „ buppOSe W= -J- W+ -j-f W + -7-77 w , then the above expression which we have to examine is equivalent to w + \dy' Uy")]-^'' dy" f(dW fdW\' /dW\"\ , 300 COMMENTATORS ON JACOBI. The coincidence of the two expressions is easily shewn hy in- tegrating {(-j-rjwdx once hy parts, and ji-r-n-j i^dx twice "by parts. Now assume that these terms of the second order can Ibe put in the form viv^ + 2v^ww' + v^w'^ + j -T-TTz {w" + \w + ixwY dx, where v, v^, v^, \, and fi are at present undetermined. In order that this transformation may be possible the following equations must be satisfied : d'V_ . ,d'V 'df~'^^^dy"^' d'V „ , ^,d'V d'V ^ . . d'V d'V _ d'V dydy''~'^''^^dy"'' d'V , , d'V dy'dy"~^^ dy'"' We may observe that if \ and /j, are known the last three of these five equations will find in succession v^, v^, and v. We do not propose to give the long process by which Spitzer solves these equations ; we will however briefly indicate the prin- ciple on which he proceeds. Let u^ denote a value of w or By which makes the unintegrated terms of the second order in the variation vanish, that is, which makes f-(f)'-(fy=o «^ SPITZEE. 301 then we may infer that this will make w' + \w + yuw vanish, that is, we shall have w/' + A.< + /iWi = (2). Similarly, let u^ be another such value of w or t>y, then Wj"+ '\u^ + fjiu^ = Q (3). Then fi'om (2) and (3) we can find X and ^ in terms of u^ and Mg and their first and second differential coefficients; and when v, v^, and Vg are expressed in terms of X and fi from the last three of the five equations given above, it remains to shew that the first two of these five equations are satisfied. When the equation (1) is developed it takes the form w""-^'2[-^w"'+\2^-^--j^-\-[-j~rT^,] + b:77^ \io" dy"^ ^"Kdy'V"" ^\dydy" dy'-^' \dydy") ' \dy"^ [dydy") [dy'^J '^ [dy'dy") (d'V /d'V\' fd'V\"\ +i^-(^+(^r=^ (^)- (d'V 1' This is a differential equation of the fourth order, so that u^ and Wg may each involve four arbitrary constants. But practically to find Wj and u^ we do not require to solve this differential equa- tion ; for we use the principle explained in Art. 252. Spitzer does shew that when \ and //- are found in the manner indicated, and then v, v^ , and v^ deduced, the first two of the five equations given above are satisfied, provided a certain relation subsists among the eight constants which occm* in u^ and u^. This agrees with Jacobi's statements in Art. 221. Since the above five equations lead by the elimination of X and fi to three differential equations of the first order for finding V, Vj, and v^, it follows that if the most general values of these quan- tities are obtained three arbitrary constants should be involved. And Spitzer shews that the eight constants which occur in u^ and u^ do combine in such a manner as to leave finally three independ- ent arbitrary constants in the values of v, v^, and v^. This gives a 302 COMMENTATOES ON JACOBI. completeness to the investigations wliich is desirable, Ibut it is not absolutely necessaiy. For all that is required in order that the proposed transformation of the terms of the second order in the variation may be effected, is that certain differential equations should be satisfied; and it would have been of no importance if the number of arbitrary constants had been less than the extreme number which the most general solutions would supply. 276. The general results at which Spitzer arrives in the in- vestigations noticed in the preceding article are the same as those of Jacobi given in Art. 224 ; but in addition to these he discusses some particular cases, and these we will now consider. Suppose then that we have -j-jj^ = ; then V must be of the form In this case the differential equation in y, which is formed by equating to zero the terms of the first order in 8 / Vdx, is a dif- ferential equation of the second order. We will suppose its inte- . gral to be y = ^ {x, a^, a^ . ^ow assume that the terms of the second order .which we have to examine can be put in the form vw^ + 2v^ww' + v^w'^ + \P{w-\- Xwf dx ; and put for shortness dy^ ~ ' dy"^ ' dy'dy" ' dW ^^ £IL,^.F. dydy" ' dydy' Then we require that Avj" + Biv'^ + Wio'w" + 2Ewio" -\-^2Fiow' SPITZER. 303 Thus we must have B=2v, + v^ + P, F= v + vl + \P. 'Now let u= C^ j^ + 0^-4^ , where C^ and C^ are arbitrary- constants ; and assume \ such that u' + \w = ; we shall then examine if we can satisfy the above five equations. The third and fourth give immediately v^ = E, v^ = D; then the second and fifth give P=B-2E-I)', v = F-E' + - (B-2E-D'); it remains to try if the values thus obtained satisfy A=v' + P\\ This requires that A = F' - E" + - (F -2E' -D") -\- '^''''" ~/\ b-2E-D') that is [A-F' ^ E") u + u {2E' + D" - B') + v!' {2E+iy -B) = 0. This equation is in fact what equation (4) of the preceding article becomes when -7-77-2 = 0, with u in the place of w ; and fi:om Art. 252 we know that the value assigned to u does satisfy it. Thus the unintegrated part of the terms which we are ex- amining, that isj JP {w -f \iof dx, becomes {{B-2E-D')(w -'^w\dx; 304 COMMENTATOES ON JACOBI. hence for a maximum or minimum 5— 2^— i)' must be respec- tively negative or positive throughout the limits of the inte- gration. Next, suppose that we have ^, = 0, and also _-2^^-^^^j =0. The unintegrated part of the terms of the second order is now and thus for a maximum or minimum A — F' + E" must he re- spectively negative or positive throughout the limits of the in- tegration. T .T ^1, .• dW fdWV /dW\" ^ . In tins case the equation -^ ( —7-7- j + f -j-^j \ = 0, is no longer a differential equation for finding w, because w"", w'", lo", and w' disappear from it. This suggests that the equation obtained by putting the terms of the first order in Bj Vdx equal to zero will not be a difierential equation in y, but an ordinary equation ; and this will be found to be the case. For Spitzer shews in the same manner as in Art. 273, that the form of V must in this case be ^ {^y y)+y'¥ («^> y) + {% {^, y, y')]\ and as in Art. 273, we shall obtain for determining y the equation # _ ^ = 0. dy dx Lastly, suppose that we have dy- "' dy- Uydy" W^W " "' and also d'^V ( d'Vy ( d''V\" + -T^ =0. d-'V f dW V / d-'V \ dy^ [dy dy') \dy dy") Spitzer shews in the same manner as in Art. 273, that the form of V must in this case be y^ {x) + [^ {x, y)]' + [x {x, y, y)]'; SPITZER. 305 . thus in SJ Vdx tlie unintegrated part is /^ {x) Bydx, and this will not vanish unless <^ {x) vanishes. Then I Vdx is exactly integrable, and its maximum or minimum can be sought by ordinary methods. 277. We may also obtain the results of the preceding article by another method. Suppose -j-jt^ = 0, then the left-hand member of equation (4) of Art. 275 takes the form {2U+ D' - B) w" + {2E' + D" - B') w' + {A-F' + E") w, where A, B, B, E, i<^have the same meaning as in Art. 276. The above expression may be written thus {{2E+B'-B)w'}'+{A-F' + E")iv, so that we have to determine the sign of ([{{2E+ B' - B) 10 ]' +{A-F' + E") v)] wdx. Suppose u such a quantity that {(2^+ B' - B) u']' + {A-F' + E") u = 0. Then the expression which we have to examine may be written j {{2E+ B' - B) w'Y - {{2E+ B' - B) u']' ^1 wdx. Integrate by parts; then the terms remaining under the integral sign will be I {B-2E-B')iv"-{B-2E-B')u'f^)' dx, that is, [{B-2E- B') (w --v^ dx. This agrees with the result at the bottom of page 303. 278. In his sixteenth section Spltzer examines some excep- tional cases which occur in finding the maximum or minimum of 20 306 COMMENTATORS ON JACOBI. / Vdx, when V involves x, y, y', y'\ and y"'. He does not here prove that V must have specific forms in certain cases, "but he assumes specific forms for V and shews that certain exceptional cases do thence arise. The following four forms for V are ex- amined. 1. {x, y, y) + y" f ix, y, y') + [% [x, y, y, y')]'- 3. j> {x, y)+y'f {x, y) + [x, {x, y, y')]' + [x, {x, y, y\ 3/")]'- 4. yj>{x)+[x,{x,y)']; + [x,(x,y:y)']: + \xz{^^y^y'^y')'^- 279. In concluding our account of Spitzer's memoirs, we may state that the most interesting and valuable portion of them con- sists in the examination of certain special cases in which the general results obtained by Jacobi require to be modified; and these special cases appear to have been examined by no other writer. 280. The next memoir we have to consider is by Otto Hesse ; it is entitled, On the criteria for the maxima and minima of single Integrals. It was published in the 54th volume of Crelle's Mathe- matical Journal in 1857, and occupies pages 227 — 273 of the volume. In the beginning of his memoir Hesse refers to the following authors who have written commentaries on Jacobi's memoir, Lebesgue, Delaunay, Bertrand, Eisenlohr and Spitzer; he makes special mention of Spitzer, and commends his acuteness and industry. It seems probable that Hesse was led to turn his attention to the subject by seeing Spitzer's investigations. 281. The first twenty pages of Hesse's memoir contain inves- tigations of Jacobi's theorems. Although there is little that is substantially new here given, the investigations are well worthy of study from their complete and systematic form. 282. In the next seven pages the result obtained by Jacobi's method is developed by the aid of tjie theory of determinants, so as to present the unintegrated part of the second variation in a more explicit form than that in which Jacobi leaves it. These HESSE. 307 seven pages constitute tlie most important portion of the memoir ; and we will give here the result obtained hj Hesse. Suppose that / Vdx is to be a maximmn or minimum, where V contains x, y^ and the differential coefficients of y up to the rif^ inclusive. Let z stand for hy ; then the terms of the second order which we have to examine can be put in the form 1 -x^ {£) zdx. This is proved by Hesse ; it is equivalent to the statement in Art. 223, that the terms of the second order can be put in the form / SVSydx. Now let u, V, w, ... be values of z which satisfy the equation '^{z) = ; we suppose n of these solutions obtained, and they will be all of the sam.G form, but differ in the values of the 2n arbitrary constants which each of them involves. Now adopting the usual notation of determinants let V, V, V , W, 10 . to" , w and V» = w, u , u , I II V, V, V , IV, w , w", An-l) „(n-a) 10'' where accents denote as usual differential coefficients. Thus v is a determinant of the {n + 1)*^ order and v„ is a determinant of the w*^ order. Then Hesse proves that the terms of the second order which we have to examine can be put into the form I N — ^ dx, J V re where N is the second differential coefficient of V with respect d-y to dx" 20—2 308 COMMENTATORS ON JACOBI. Moreover Hesse draws particular attention to the fact that certain relations must hold among the arbitrary constants involved iau,v,w,... See Arts. 221 and 232. 283. The remainder of Hesse's memoir is devoted to the examination of three particular cases, that in which the integral involves x, 7/, y', that in which the integral involves x, y, y', y", and that in which the integral involves x, y, y' , y" ., y" . These cases are treated very fully, and the relations which hold among the arbitrary constants are completely exhibited. No notice however is taken of the exceptions to the general theory which Spitzer considered; see Arts. 273, 274, 276. In connexion with the first of the three particular cases which he examines, Hesse gives a good discussion of the remarks made by Jacobi relating to the extreme limits which may be assigned to an integral in order to ensure a maximum or a minimum ; see Art. 225. 284. The memoir by Hesse forms the most elaborate commen- tary that has yet appeared on Jacobi's theorems and method. The student who masters this and examines what Spitzer has given on the exceptional cases will not require any further information on the maxima and minima of single integrals which involve one dependent variable. Hesse uses the theory of determinants, but a student who is acquainted with the elements of that subject will not find any serious difficulty in Hesse's memoir. 285. We have next to consider a memoir by A. Clebsch ; it is entitled On the reduction of the second variation to its simplest form. It was published in the 55th volume of CreUe's Mathemati- cal Journal in 1858, and occupies pages 254 — 273 of the volume. This is the first of three memoirs by this writer on the Calculus of Variations. He begins by referring to Jacobi's results, and to the excellent memoir published by Hesse respecting them. He then indicates the points in which Jacobi's results require still to be generalised, namely, that similar investigations should be sup- plied for the case of a single integral which involves more than one dependent variable, and for the case of a multiple integral, and for the case in which equations are given connecting the variables involved in the integral. The present memoir proposes to supply CLEBSCH. 309 some of these required investigations. Thus, Clebsch states that the memoir solves the following problem ; to reduce the second variation of a single integral so as to make it depend upon the smallest number of variations, the integral involving any number of dependent variables and their differential coefficients to any order, and also any number of equations being given connecting the variables. In addition to the solution of this problem, there is a short section on the subject of multiple integrals, but this is of no great importance; the Avriter however intimates that at the time of printing he had succeeded in overcoming the difficulties of this part of the subject, and would publish a memoir on it, and in fact the third memoir fulfils this promise. As an example of his method, Clebsch gives the ordinary case of one dependent variable without any connecting equations, and he arrives at the result obtained by Hesse; see Art. 281. 286. The second memoir by Clebsch is entitled On those problems in the Calculus of Variations which involve only one in- dependent variable. It was published in the 55th volume of Crelle's Mathematical Journal in 1858, and occupies pages 335 — 355 of the volume. This memoir may be said to consist of two parts. The first part is occupied in proving that the solution of any problem in the Calculus of Variations in which there is only one independent variable may be made to depend on the solution of a certain partial differential equation of the first order. It had been intimated by Jacobi that this proposition was true in a certain case, so that a problem in the Calculus of Variations could be treated in a manner analogous to the treatment of dynamical problems by the methods of Hamilton and Jacobi. Clebsch easily proves the proposition which he enunciates. The second part of the memoir is occupied in shewing that this mode of treating a problem in the Calculus of Variations presents great advantages in the discussion of the terms of the second order with the view of discriminating between maxima and minima values. This part of the memoir is extremely complicated and 310 COMMENTATORS ON JACOBI. requires the reader to possess a good knowledge of tlie theory of determinants. 287. The third memoir bj Clehsch is entitled On the second variation of Multiple Integrals. It was published in the 56th volume of Crelle's Mathematical Journal in 1859, and occupies pages 122 — 148 of the volume. The object of the memoir is to shew how to discriminate be- tween the maxima and minima values of multiple integrals ; like the second memoir by the author it is extremely ^complicated, and requires the reader to possess a good knowledge of the theory of determinants. 288. We have been compelled to give very brief accounts of the memoirs by Hesse and Clebsch. From the nature of the memoirs it seems impossible to present any abridgement of them or any extract from them which will be easily intelligible ; and moreover the memoirs belong rather to the Theory of Determinants than to the Calculus of Variations. As however these memoirs have been published so recently they can be readily obtained, and thus there is less need of a detailed account of them than in the case of works which are more difficult of access. CHAPTER XL ON JACOBI'S MEMOIR. 289. The preceding chapter contains an account in chrono- logical order of the writings of commentators on Jacobi's memoir ; the present chapter consists of some miscellaneous articles bearing on certain parts of Jacobi's memoir. In Art. 228 we have given Jacobi's remarks on the shortest line that can be drawn on a surface ; these remarks are connected with those in Art. 225. We have also intimated that some of the commentators on Jacobi's memoir have considered these parts of it ; see Arts. 265 and 283. There is a note by J. Bertrand entitled On the shortest distance hetioeen two points on a surface, which was published in 1855 in the second volume of the third edition of Lagrange's Mecanique Analytique, pages 350 — 352. We will give this note in the next article. 290. When a material point moves on a fixed surface, and has an initial velocity but is acted on by no force, Lagrange proves that its velocity is constant and tlie curve which it describes is the shortest that can be drawn between two of its points. In order to prove this proposition the illustrious author shews that the varia- tion of the arc j ds is zero, and therefore there is either a maximum or a minimum ; but he says there cannot be a maximum and there- fore there must be a minimum. This manner of reasoning is in- admissible, because we know that the variation of an integral may be zero while the integral is neither a maximum nor a minimum. However in the particular case in question Lagrange's statement is exact, as we may shew in a few words. 312 ON JACOBl's MEMOIR. The differential equation whicli expresses tliat the variation of the integral / ds is zero proves, as is well known, that the oscu- lating plane of the cnrve is at every point normal to the surface. But if we suppose the two extremities of the arc considered to he indefinitely close, among all the arcs drawn on the surface joining the extremities, the least, that in fact which differs least from the chord, will evidently be the arc which has the least curvature, that is, the arc which has the greatest radius of curvature. But the arcs which unite two points of the surface indefinitely close may he considered as having the same tangent, and therefore, by the well-known theorem of Meunier, that of which the osculating plane is normal to the surface has the greatest radius of curvature, and is consequently the shortest. The proposition enunciated by Lagrange is exact as we have just seen for any indefinitely small arc, but it would cease to be so if we considered an arc of finite size. There exists a curious theorem on this subject enunciated by Jacobi without demonstra- tion, which gives a general method for determining with respect to every line traced upon a surface and satisfying the conditions of a minimum, the limits between which it is really the shortest line. Let AMA' he such a line ; jproceed along this line from the ^oint A which is fixed towards the following ^points *of the curve. If we take one of these points as a second limit it tnay hapjpen that between this point and the first another curve can he drawn which satisfies the analytical condition for a minimum as well as the first; then the line considered will cease to he a minimum between the point A and the second extremity considered, at a point for which the second line coincides loith (se confond) the first. This theorem has not been demonstrated by the mathematicians who have explained the celebrated letter in which it is enunciated. We think that it will be useful to indicate briefly how it follows from the analysis of Jacobi. The integral considered being \flx, y, -^jdx, the variation takes the form VBydx, V being the function which by being BEETEAND. 313 equated to zero furnishes Ibj integration the solution of the pro- Hem. That there may be a minimum, the function -t4^ must remain constantly positive during the limits of integration. This con- dition -will certainly be fulfilled whatever the limits may be, be- cause we have seen that there is always a minimum between any two limits whatever if they are sufficiently close. Besides this we must have according to Jacobi's analysis another condition ful- filled. Let y denote the expression deduced from the equation V= 0, then y contains two constants a and h suppose ; let a and /S be two other constants, and let Then the other condition is that it must be possible to take the constants a and /3 so that the expression / dj 1 d\fdu \dydy' u dy'^ dx. may not become infinite between the limits of integration, or, which comes to the same thing, u must not vanish between these limits. Hence it is clear that for each value of — if the expression for u becomes zero in two points of the minimum line furnished by the Calculus of Variations, then between these two points we can affirm that the integral is a minimum. Now two such points will possess the property indicated by Jacobi, that is, it will be possible to draw between them two lines indefinitely close, each of which has the minimum property. For we observe that the expression d^i r, dy da do J is the general integral of the linear equation SV=0, in which Sy is the unknown quantity. (See Jacobi's Memoir.) If then we put instead of y the value y + u, where a and /3 are so chosen -as to make u indefinitely small, which is allowable, the expression V 314 " ON JACOBl's MEJMOIR. will vanish, "because by supposition y makes it vanish and the indefinitely small increment u renders its variation zero. Thus there are two lines indefinitely close joining the same two points, for which the relation V= is fulfilled, that is, two lines which equally satisfy the conditions of minimum. The proposition thus demonstrated is not identical with that of Jacobi, but it is perhaps allowable to suppose that the illustrious author went a little too far in the rapid sketch which he gave of his results ; it is clear, for example, that the conditions found by him are sufficient but not necessary for the existence of a minimum. There is therefore no ground for affirming that the minimum ceases to exist because the function u becomes zero ; but this would be necessary in order that the enunciation should take the completely affirmative form given above. We may observe before closing this note that Jacobi' s memoir contains the enunciation of another very remarkable theorem ; if at every point of a surface the two curvatures are in opj^osite directions the line which satisfies the analytical conditions of a minimum is always really the shortest. We confine ourselves to recalling this theorem to the attention of mathematicians ; a more detailed dis- cussion of the geometrical problem which is the object of this note would be out of place here. 291. Bertrand in the first paragraph of his note says that Lagrange is right in asserting that there is necessarily a minimum in the case considered ; in the third paragraph of his note he says that Lagrange's statement is not necessarily exact except for an indefinitely small arc. The remarks which Bertrand then makes on Jacobi's theorem coincide in substance with those of other writers on this subject ; see for example Mr Jellett's treatise, pages 90 and 98. These remarks depend on the following consideration ; Jacobi's method reduces the unintegrated part of the terms of the second order to the form 2]dy"[u dx -^ dx) ' and thus we cannot be sure of a minimum if it vanishes between the limits of integration. Bertrand however is alone in pointing out that this does not prove so much as Jacobi asserts in the BONNET. 315 particular problem tinder consideration, tor Jacolbi asserts tliat there ' will not be a minimum ; and Bertrand conjectures that Jacobi here overstated his results. But it has since been shewn bj Ossian Bonnet that Jacobi was quite correct; the proposition to which Bertrand calls attention at the end of his note is also proved by Bonnet. Two notes have been written bj Bonnet on the point we are considering. The first note is entitled On some properties of geo- desic lines. It was published in the Comptes Rendus de VAcademie des Sciences, Vol. 40, 1855, pages 1311 — 1313. "We will give it in the next article. 292. A line traced upon a sm'face is called a geodesic line, when its osculating plane is always normal to the surface. An arc of a geodesic line is the shortest line that can be drawn on a curved surface between its two extremities, provided the arc be comprised within certain limits which have been fixed by Jacobi in the following manner. Consider a geodesic line AM which starts from the point A, and let A' be the point where this line is met by another geodesic line AM' which also starts from the point A and is indefinitely close to AM. Between the points A and A' the line AM will always be a minimum line ; but beyond the point A' the line AM will generally be neither a maximum nor a minimum. Assuming this, let p denote the variable distance MM' between two indefinitely close geodesic lines AM and AM'. By a formula due to Gauss, p) considered as a function of the arc AM will satisfy the difierential equation of the second order ds^ ^ RE ' where AM= s, and R, R are the principal radii of curvature of the surface, and in addition, when s = we have p = Q and -~ = the angle d6 between the geodesic lines AM and AM', which will completely determine p. Now suppose that the surface is of opposite curvatures ; -5-57 will be negative, and we can assume 1 _ 1^ RR' ^ ce where a is a real constant. 316 ON JACOBl'S MEMOIE. Let us take the equation d^P. _P. _ ft ds" a' ' and integrate it so that when s = we may have /?, = and -^ = d6; we shall obtain as a , - But from a theorem demonstrated by M. Sturm in his excellent Memoir on differential equations of the second order, it is known, that for any interval whatever starting from s = 0, the value of ;p, must vanish at least as often as that of -p ; but p^ never does vanish, and so p cannot vanish. Thus in a surface of opposite curvatures a geodesic line is always a minimum throughout its length. This beautiful theorem was enunciated by Jacobi, but it had not been demonstrated up to the present time so far as I know. Suppose in the next place that -^^ is positive and less than -^ . Consider the equation di ^ a' ^' df) and integrate it so that when s = we may have p, =0 and -4-' = dO ; we shall obtain s p^ — adO sin a But, from a second theorem demonstrated by M. Sturm, it is known that, starting from s == 0, ^ will vanish before p^ ; but p^ vanishes when s = ira, therefore p vanishes before s = tra. Hence we infer that, in the case considered, a geodesic line cannot be generally a minimum line throughout a greater length than ira. Consequently the shortest distance between any two points on a convex surface is less than ira where ,a^ is a number greater than the product B.B! of the principal radii of curvature for all points of the surface. BONNET. 317 293. The theorem due to Gauss which Bonnet quotes in the preceding article is contained in the Disquisitiones generates circa superficies curvas. This memoir was presented by Gauss to the Eoyal Society of Gottingen on October 8th, 1827, and was published in 1828 in the sixth volume of the Commentationes Recentiores of that Society. This memoir is reprinted in Liouville's edition of Monge's Application de V Analyse a la Oeoinetrie. The theorem in question is also proved by Ossian Bonnet in his memoir on the general theory of surfaces in the Journal de VEcole Poly technique, Cahier xxxil, 1848. The memoir by Sturm to which Bonnet refers will be found in the first volume of Liouville's Journal of Mathematics. The second note by Bonnet is entitled Second note on geodesic lines. It was published in the Comptes Bendus ...Yo\. 4,1, 1855, pages 32 — 35. We give it in the next article. 294. In a note presented to the Academy on the 18 th of June, I established some general properties of geodesic lines. My in- vestigations depended on the following theorem due to Jacobi. Let AM he any geodesic line which starts from the point A, and suppose A' to he the point where this geodesic line is met hy a geodesic line which also starts from the point A and is indefinitely near to the first; then the linB AM will he a minimum hetween the points A and A' , and will cease to he a minimum heyond the point A'. Jacobi did not demonstrate his theorem ; he merely said that it might be easily deduced from the general rules which he gave for distinguishing maxima from minima in questions which de- pend upon the Calculus of Variations. M. Bertrand has given a proof of the first part of the theorem in the notes which he has added to his excellent edition of the Mecanique Analytique; that is, he has proved that between the points A and A the line AM is a minimum. In the mode of proof M. Bertrand has followed the indications of Jacobi. With respect to the second part of the theorem M. Bertrand thinks that it may not be exact, and that at all events the method of Jacobi is not competent to decide the point. It is in fact certain that the general conditions found by Legendre and completed by Jacobi, for distinguishing between 318 ON JACOBl'S MEMOIR. maxima and minima in problems which depend upon the calculus of variations are sufficient hut not necessary. I have succeeded in proving by particular considerations both parts of Jacobi's theorem. I request permission from the Academy to communicate my demonstration, which thus removes the difficulties which re- late to an important question, and at the same time gives more precision to the results of my previous investigations. Let AMB be an arc of a geodesic line which starts from A and ends at B. (The reader is requested to make the figure for himself.) Draw any line AM^B indefinitely close to AMB and having the same extremities. I proceed to estimate the difference of the lengths of AMB and AM^B as far as small quantities of the second order ; for this purpose I draw through the different points of AMB geodesic curves normal to AMB, and I denote in general by ft) the portion of these curves comprised between AMB and AM^B. Suppose the element MN of AMB = ds, and the con-e- sponding element M^N^ of AM^B = ds^ ; then M^N, = ds^.= ^[{M,PY + {PNJ% P being the point in NN^ such that NP = MM^ . But PN. = -^ ds = fo'ds, ^ ds and M^P is, by a theorem due to Gauss, the integral of the equation d'^u u dc^'^BB'^ ' which for w = satisfies the conditions zi = ds, ^- = 0. Therefore dco if we neglect powers of w above the second. Therefore or more simply, to the order of approximation which we want, ds=dsil+\<.''-\-^). BONNET. 319 Therefore the difference between AMjB and AMB, that is the second variation of the integral jds, will be - \\oi---^,]ds (1). 2jr RR' We see immediatelj that if RR' be negative this second varia- tion is always positive ; this proves the first theorem which I have established in another manner in my first note ; in any surface of opposite curvatures a geodesic line is a oninimum throughout its length. Now let lis call p the distance comprised between the line AMB and another geodesic line indefinitely close to it which also starts from A, so that we have ds'^RE ' and when 5 = we have ^ = and -— = the indefinitely small angle dd between the two geodesic lines ; the expression (1) can be put in the form But, if p does not vanish within the limits of integration, for 0) is zero at the limits ; thus the second variation is reduced to ISi-'-t^h that is, to a positive result. I conclude therefore, .that so long as the extremity B is not beyond the point A' where the line AMB is met by the geodesic line indefinitely close to it which also starts from the point A, the arc AMB of the geodesic line is a minimuni between the point A and the point B ; this is the first part of Jacobi's theorem. 320 ox JACOBI'S MEMOIR. If tlie point B is beyond A', then, since « is only subject to the condition of vanishing at the points A and B, we can take for to a value which satisfies an equation of the form . where h is real, and which is such that « = and -^ =d6 when 5=0. This follows from the fact that in an equation of the form when G is diminished continuously the roots of the equation p = increase continuously, (p and -^ retaining the same values for s=0). We have then for this particular value of to „ ft) ft) but, since o) is zero at the limits, we have also ft)^ BR' (o'^ — -^pTri, ] ds = — Ift) [ ft)" + -^-^ j ds ; therefore /(""-w)''^ =-/?*• Thus the second variation of the integral fds can become nega- tive, and the arc AMB is neither a maximum nor a minimum between the point A and the point B, The second part of Ja- cobi's Theorem is thus established. We have said above that when once Jacobi's Theorem is fully demonstrated we can give more precision to the enunciation of the results contained in the note of the 18th of June. In fact we can say that if in any convex surface the product RR' of the principal radii of curvature is less than the constant a^, the shortest distance from one point to another upon the sm-face will always be less than ira. Hence it follows that evory convex sm-face in which the principal radii of curvature do not become infinite is neces- sarily a closed surface. HEINE. 321 295. Althougli so many proofs have been given of Jacobi's theorems that it may appear superfluous to present others, yet the following proofs are of interest as they depend on the principles of the Calculus of Variations itself. They were published in an article entitled Observations on Jacdhis Memoir on the Calculus of Variations, by E. Heine, in Crelle's Mathematical Jom-nal, Vol. 54, 1857, pages 68 — 71. They will occupy our next two articles. 296. The proposition which Jacobi published in the 17th volume of Crelle's Journal and which was proved by Lebesgue and by Delaunay in the 6th volume of Liouville's Journal may be demonstrated also, without much trouble, in the following way, which depends on very different principles. Let A be any given function of a?, u any function of a?, and let w, m", ... denote the difierential coefficients of u with respect to X. Put 2^= (- lYJAu^^^u^^^dx (1), where w'"' stands for -y-^ ; then '=(-1)"/ Now by integrating by parts in the ordinary way SZ can be separated into two portions, namely, one which is free from the in- tegral sign, and which we will call L, and another portion which remains under the integral sign, namely. / — V-;; — ~ buax. Let y denote any given function of x, and put u=yt, so that tu=yht'j thus ^Z = L+\y^^^^^htdx (2). Now we must obtain an equivalent value for hZ if we first put yt for u in (1) , and then effect the variation ; the form how- ever of the expression for hZ will differ from that in (2) ; and this difference in form accompanied with equivalence of value will give a proof of Jacobi's Theorem. 21 322 ON JACOBI'S MEMOIR. Put yt for u, tlien %'"' takes the form a«'"' + a,«'"-'' + . . . + oinj + a,/, where a, a^, ... are simple functions of y and therefore functions of X. Thus -4m'"' m'"' will consist of a series which we may denote by S/3<""' ^*^' where the indices m and p may take all values between and 71, and the functions denoted by ^ will be like a, a^ , . . . , given functions of x. Put this expression for ^m*"'m'"' in (1), then we shall shew that 2Z can be put in the form 2Z=M->t hc^tt- C^t't'+ Cft"-...±CJ''H^''')dx (3), where M contains no integral sign, and C^, C^, ... are given func- tions of X. For S i ^t^^H'^'^ dx consists partly of terms for which m =j>, which thus have abeady the form in (3), and partly of terms in which m and p are different. Suppose then p greater than m, and first let p = m+ 1; for such terms l^t^^^H'^^dx^l^r^r^'^dx a Am) Am) /• ly and thus we obtain again terms of the form in (3). Next suppose p — m greater than unity ; then by using the following formula hrk^^^dx = ^rH^^^- fov""'^^'' dx - l/3i""+^'^<^-^' dx, as often as necessary, we shall obtain terms in which the indices of t are either equal or differ by unity ; and thus finally we obtain terms of the form in (3). Now take the variation of Z expressed as in (3) ; then by the ordinary formulae of the Calculus of Variations the term in hZ which remains under the integral sign is l\-^-^-'W--^'W^^'- «• HEINE. 323 The expression (4) must therefore be equal to the integral in (2) ; thus The quantity G^ is equal to y — V#-^ . For since u = yt, we have thus the term d"" {Ay^""') y dx"" is the only term which can contribute any portion to G^t, and thus obviously We can now prove Jacobi's Theorem. Let C7=^„. + ^^^ + ...+-i^, where ^„, A^^ ... are given functions of x. Put u=yt where ?/ is a given function of x ; then from what has been proved y^-^^^^—d^^ + dx- ' where B^, B^, ... are known functions of a?, like Co, G^, ... were. Also B-Ja.a-^^^^^a- +^1M^I. Also i^^_3/|^„2, + -^-_ + ...+ ^^„ |, thus when y is so chosen that it is an integral of the differential equation U=0, we have ^^ = ; and then r ^, ^ , d(Bj") d^-'iBJ""^) as Jacobi's Theorem asserts. Itemarh. In order practically to determine the values of B^, B^, ... which do not come into consideration in Jacobi's memoir, 21—2 324 ON JACOBI'S MEMOIR. the method may be modified by first integrating by parts and thus reducing lAu^''^u^''^dx to \u \ „ — ' dx. Now put f^for u, and then we have to consider integrals of the form Ifitt^^^dx, and not as be- fore integrals of the form i^f'H^^dx. 297. Jacobi published another proposition in his Memoir, of which Delaunay has given a long demonstration in the place already named. This is the proposition ; let J=jfix,y,y',...yn<^^, then BJ consists of a part free from the integral sign together with the integral I VBy dx, where ^ ^ ^y' dx ^ - dx"" This is well known ; then Jacobi asserts that 8 V may be put in the form A^^'J^^...^'^S^ = W. (6). We proceed to prove this. Let 8 and 6 be sjTnbols of variation which are independent of each other j then the double variation 8 6 J wiU be equal to s//"(y"',^"") (8y'"'%^>+ %'™'8y^') dx (7), where the sign of summation refers to all values of m aud^ which are comprised between and n. But on the one hand this expres- sion must be of the form L-^hvOydx; for if we vary / with the symbol 6 we should obtain an integrated part and the unintegrated part I Vdy dx ; and if we now vary the result with the symbol 8 we obtain for the unintegrated part hveydx. HEINE. 325 Moreover the expression given above for 8^/ can be put in the form M+l{AByey - A^ydy + . . . + ^„Sy"'%'"') dx (8), as we shall shew presently. Now by the ordinary method of integrating by parts the unintegrated part of the last expression is found to be \W0ydx; and thus BV=W, which was to be proved. We have then only to shew that 8 0J really has the form (8). The terms in (7) for which m =^ have already the required form ; suppose then^ greater than m, and first let p = m + l. Then it is plain that by single integration h (S«/""'%'"^-^« + S3/"««'%<'"') dx is referred to the form I ^'By^^'Wy^""^ dx. Kj) be greater than m+l, then by single integration we make h (S?/""'^^'^' + By^Wy"^') dx depend on and f /3' (82/""'%'^" + %'^"%'«") dx; and by proceeding thus we shall ultimately arrive at the form in (8). 298. There is an article by Minding entitled. On the trans- formations which serve for distinguishing maxima from minima in the Calculus of Variations. It was published in Crelle's Mathe- matical Journal, Vol. 55. 1858, pages 300 — 309. The object of 326 ON JACOBI'S MEMOm. this article is to demonstrate two theorems used in Jacobi's memoir ; namely, the theorem in Art. 222 in the form in which it is given in Arts. 229 and 231, and the theorem respecting the form of SF in Art. 223. The demonstrations are somewhat complex, but per- fectly satisfactory; as they consist however almost entu-ely of ordinary algebraical transformations it will be unnecessary to enter upon them here. 299. We will close this chapter by giving two examples of the investigation of a maximum or minimum value. For the first example we will apply Jacobi's method to the expression I Vdx where V=(p + ^] x-\- {2c + cif) x. This is in fact the example given on page 108 ; the quantities which were there denoted by r, ^, -^ are now denoted respectively by a?, y, p. The expression of the second order which determines whether there is a maximum or a minimum is here f L [hpY + 2SpS^ + (ex + ^) %)4 clx. Let ^^cxy-x^^(p + y^, therefore S/3 = cxhy — a; -j- [ S^? + — j ; therefore I B^Bydx = I ex {Byf dx — \ xSy y [ Sj> + — j c?a; = -xBy(Bp+^)+jcx{SyYdx+j(Bp+^-l)^ixBy)dx =:-xBy(Bp + ^-^^+ji^x{B2>y+2B2?By+(cx + ^ {ByJ^dx. Thus we see, since the limits are supposed fixed, that the terms which we have to examine can be put in the form JB^Bydx; this is in accordance with Jacobi's theory. AN EXAMPLE. 327 Now \ B^S7/dx=\ uB/B — dx, where u is at present undeter- mined ; also if u be properly determined we shall have for this only requires that d f dBi/ -> die] = — 7- iux-^ — xby -J- y , dx { ax "^ dx) d / dBy\ dBy du d I ^ du\ ^ "~ . dx\ dx j dx dx dx\ "^ dx) ' that is, we must have r ]^^^( ^\ y. xj dx\ dx) ' Suppose then that u is taken to satisfy this differential equation ; then we get JBl3Sydx=ju'x\i-(^)[dx: [dx V u . neglecting the terms free from the integral sign, which vanish at the limits if no infinite quantities occur. Now u is such a quantity that if put for By in B^ we get Sy8 = 0; hence the value of u is known by Jacobi's theory; see Art. 220. The value of y which makes B I Vdx = is in the dz present case to be obtained by finding -j- from the value of z on page 108, and then changing r into x. Thus it is dx J Q ux J Q and therefore the value of u is in the present case of the same form, with A and B replaced by new constants. The second constant r 328 ON JACOBI'S MEMOIR. must be supposed zero in order that u may not be infinite when a? = ; hence finally r e-*^"°''"cosG)c?w, where a is an arbitrary constant. This value of u however vanishes when x = 0, so that the ex- pression under the integral sign in the value of I h^hydx becomes infinite when x = Q. Hence we are not certain that in this case we really have obtained a minimum. 300. The next example is intended to draw attention to the case in which we have to discriminate between the maximum and minimum of a function when the limits are not fixed. Writers on the calculus of variations appear frequently to intimate that the fact of the limits being variable does not really render the problem more difficult ; this however does not seem correct. Let us consider the problem of the brachistochrone in which the moving particle is to pass from one given curve to another, starting with an assigned velocity. Take the axis of x vertically downwards ; let h be the height due to the initial velocity, a?^ and x^ the abscissse of the starting-point and the final point respectively. Then we have to find the minimum value of I Vdx, where . - J Xi V= ,,}• . — ^-^ and r) = -f-. We shall treat the problem in hJ\h-\-x — x^ '■ax ^ , what seems the best way ; we shall attribute no variation to the independent variable x but shall obtain the requisite generality in our formulse by changing the limits of the integration. Suppose then that p receives the variation 8^, and that the limits x^ and x^ become respectively x^ + dx^ and x^ + dx^. In consequence of the change in^ and x^ a change takes place in F, and to the second order V becomes AN EXAMPLE. 329 Hence tlie variation of tlie integral is Vdx- Vdx J Xi+ dxi J Xi the limits in the last line being x^ + dx^ and x^ + dx^. Now we observe that if in an integral I ^ (x) dx the upper limit is increased by dx^ the integral is increased by to the second order ; and if the lower limit is increased by dx^ the integral is diminished by dx,{x;) + -{dx;)^'{x;}, to the second order. Thus the above variation becomes to the second order F,«fe, - F, second order, The interpretation of the terms of the first order is well known, tut we will give it here to render our investigation complete. Equate to zero the coefficient of dx^ ; thus {dV ,,, , dV dy ^J ^ dV substitute the values of Fand -^ , and we obtain f pyjr' (x) + 1 W{l+f)^/{h + x-x;) Thus {py^' {x) + l}^ = 0, which shews that the curve described cuts the lower limiting curve at right angles. Next equate to zero the coefficient of dx^; thus JNow -7— = ^^ ^ 8 , and by supposition -^ is equal to a constant, that is -jj- „, f,, r = -7- say ; ' /v/(i+i?) V(^ + «-«i) V« hence p = ^ — , 1 + » = 5 . ■^ a — h — x + Xj^ *■ a— h — x + x^ mi dV fJa ihus -5— = ^! — I =: «^i 2{a-h-x+x^)i{h + x-x^)^ 332 ON JACOBl's MEMOIR. and i^^dx- ^fjci-fi-x+x,) ^ 1 J dx^ ^/a '\/{h + £c — fljj p \/a' Hence / ^■^f—dx = 7- f ). J x^ dx^ sja Vpa VJ Thus our equation "becomes 1 that is ^4:^+ 1 + 1 fi_i)=0. Therefore %' {x^'p^ + 1=0, and thus the tangents to the limiting curves at the points where the described curve meets them are parallel. We have now remaining in the variation only terms of the second order ; by reduction they become + \ J2 {?(8P)'+ 2 1^ Bp rf.. + ^^ (i..) j d.. Now it is by no means evident that the above expression is necessarily positive, so that we are not sure of the existence of a minimum as asserted by Legendre ; see Art. 203. Nor do Jacobi's investigations give us here any assistance. The above expression shews that cceferis paribus the suppositions that i/r" (x^ is positive and that x' i^i) ^^ negative are favom*able to the existence of a minimum. This makes the lower 'limiting curve convex to the axis of X and the upper limiting curve concave to the axis of x at the points where the described curve respectively meets them ; and it is obvious from a figure that these circumstances are favom-able to the existence of a minimum. CHAPTER XII. MISCELLANEOUS MEMOIKS. 301. The present chapter contains an account in chronological order of various articles, memoirs, and treatises, connected with the Calculus of Variations. 302. Poisson, Memoires de Vlnstitut, 1812, page 224. Poisson here finds the differential equation to the surface of con- stant area which makes 1 1 V(l +P^ + 2'^) (~ + ~ ] dxdy 2^ minimum, where p and p are the principal radii of curvature at the point {x,y, z) of the surface. He adds that the equation obtained would also be obtained if we required that / /VCl +I>'^ + 2^) ( /) ^^dy should be a minimum, or that \\'\/{)-+p^ + ^){-i + —zjdxdy should be a minimum ; for I \h— —, — — dx dy vanishes, so far as the terms under the sign of double integration are concerned. There are two misprints in Poisson's remarks, but there can be no doubt that his meaning is what we have here given. 303. Eodrigue. Bulletin des Sciences 'par la Societe PMlo- matigue de Paris, 1815, pages 34 — 36. This paper is on certain properties of double integrals and of the radii of curvature of surfaces. It is stated that the variation of the double integral jj{p, q) [rt — s^) dxdy contains only terms 334 POISSON. CHOISY. relative to the limits. This may be verified without much diffi- culty ; that is, we can shew that the part of the variation under the double integral sign is identically zero. Hence we see that this statement is an extension of that quoted in the preceding article from Poisson. 304. Poisson. Bulletin des Sciences par la Societe Fhiloma- tique de Paris, 1816, pages 82 — 86. This paper is on the Calculus of Variations with respect to multiple integrals. Poisson refers to the difficulty which Lacroix had found in the variation of a double integral, which led him to infer that hx must be supposed a function of x only and hy a func- tion of y only ; see Art. 40. Lagrange adopted the same hypothesis as sufficient for his purpose without asserting its necessity ; see Mecanique Analytigue, 3rd edition. Vol. i. page 92. Poisson re- moves the difficulty by giving the correct expressions for Ss, hz^, ... instead of those given by Lacroix. The substance of this paper was given by Lacroix in his third volume, pages 717 — 720; and it was afterwards incorporated by Poisson in his memoir on the Cal- culus of Variations. See Art. 102. 305. Choisy. Essai Historique sur le prohleme des maximums et minimums et sur ses applications a la mecanique par J. D. Choisy, Geneva, 1823. This work consists of 66 quarto pages. It is divided into two parts. The first part is on the abstract problem of maxima and minima ; this contains five chapters; (1) Preliminary considerations, (2) Elementary and synthetical methods, (3) Analytical methods up to those of the Bernouillis inclusive, (4) Methods of Euler, (5) Methods of Lagrange. The second part is on the applications of the theory of maxima and minima to Mechanics ; this contains six chapters ; (1) On the use of indeterminate coefficients in the appli- cations of the Calculus of Variations to Mechanics, (2) On the principle of least action, (3) On the Cycloid, (4) On the Catenary, (5) On elastic curves, (6) On equilibrium. At the end of the work is a list of authors on the subject ; this list does not seem to contain anything of importance in addition to the usual references. GRAEFFE. 335 The present writer has never seen Choisy's work ; for the albove notice of it he is indebted to a friend, who at his request examined the copy in the Bodleian Library at Oxford. 306. C. H. Graeffe. Commentatio Historiam Calculi Varia- tionum inde ah origine Calculi Differ cntialis atque Integralis usque, ad nostra tonpora complectens. This essay obtained a prize fr'om the University of Gottingen in 1825 ; the adjudicators however state that it is defective in giving so little information on the more recent investigations re- lating to the Calculus of Variations. The author in his preface states his intention of going farther into the subject in a future essay ; this intention however does not appear to have been ever carried out. The essay occupies 60 quarto pages ; it ^*aces the history of the subject from its origin until the time of Lagrange. The essay thus goes over the same ground as the well-known work of Woodhouse. It is however not so full as the work of Woodhouse ; it sometimes merely states that certain results were obtained, without explaining the method by which they were obtained. The essay does not bear upon the subject of the present volume, because it scarcely alludes to anything after the works of La- grange. A few lines ai'e given to Dirksen, a few to Ohm, and a few to Buquoy ; the latter two are not highly estimated by Graeffe. Thus he says : " Conatus quos Ohm ad hunc calculum stabiliendum publicavit parvi momenti sunt...," and "... ad calculi variationum principia fundanda Comitem de Buquoy etiam, quanquam frustra, vires tentasse ; non est tamen meum propositum hos conatus scien- tiam non augentes accurate explicare." Graeffe refers to Lacroix in the following terms: "...inter eos qui libros quibus doctrines matheseos exponuntur perscripserunt, Lacroix calculum variationum diligentissime tractasse." These extracts are all taken from the last two pages of Graeffe's work. The present writer has never seen the work of Buquoy to which Graeffe refers ; its title appears to be Eine eigene Darstellung der Grundlehren der Variations-i^ecTinung, and the date 1812 is ascribed to it in a bookseller's catalogue. 336 MINDING. 307. Minding. Grelle^s Mathematical Journal, Vol. 5, pages 297 — 304, 1830. This article is entitled On curves of shortest perimeter on curved surfaces; it contains a discussion of a problem proposed in tlie third volume of Crelle's Journal by Crelle himself. The problem is to find the shortest curve which can be drawn on a given surface so as to include a given area. Minding obtains the following results. If the given surface be a sphere the required curve is a plane curve, and therefore a circle. He obtains the re- quired curve when the surface is a right cone. He remarks that if the surface be any developable surface, the required curve must be such as will become a circle when the surface is developed; this follows from the known fact that of all plane figures a circle is that of least perimeter which bounds a given plane area. Minding also establishes the following result. Whatever be the surface the curve required has this property ; the cosine of the angle between the osculating plane of the curve at any point and the tangent plane of the surface at that point is proportional to the radius of curvature of the curve at that point. This property has since been proved by other writers who have discussed the problem, namely, Delaunay, Bonnet, Jellett, and Schellbach. The last five pages of the article are occupied with an investi-^ gation respecting another property of the curve ; Minding appears to have here fallen into an error, and some detail will be required to illustrate the point. A geodesic line is a curve drawn on a surface so that at every point its osculating plane contains the normal to the surface at that point. Now suppose a series of geodesic lines starting from a common point on a surface, and let a series of cm-ves be drawn cutting these geodesic lines at right angles. The latter curves may be called geodesic circles, because it can be proved that the length of the geodesic line drawn from the common starting-point to any point of one of these curves is constant. This property of a geodesic circle from which its name is derived is proved by Minding, although he does not use this name. The name is used in Price's Infinitesimal Calculus, Vol. II. and the pro- perty is there proved; see also Bonnet's Memoir on the general MINDING. 337 theory of surfaces in the Journal de VEcole Polytechnique, Cahier 32, page 74. The property then which Minding considers that he proves is that the curve of least perimeter which can be drawn on a given surface so as to include a given area is a geodesic circle. This is in fact true for any developable surface in virtue of the remark already made ; but it does not appear to be generally true. It is however remarkable that Bonnet and Schellbach, who both seem to allude to Minding's solution, take no notice of this part of it. We will indicate the grounds for considering this part of Minding's article to be erroneous. Let p be the radius of curvature at any point of the required curve, 6 the angle which the osculating plane at any point of the curve makes with the tangent plane to the surface at that point. Then the characteristic property of the required curve is that = a constant. If then Minding's result were correct it would follow that this property must necessarily belong to a geodesic circle. Suppose, for example, that we consider an ellipsoid ; let the semiaxes be a, b, c in descending order of mag- nitude ; and suppose we require the cm-ve of least perimeter which can be drawn on the sm'face so as to enclose an area equal to half that of the ellipsoid. It would appear obvious that the required curve must in this case be the ellipse which has h and c for its semiaxes ; for this curve satisfies the condition = a constant, P since cos ^ = 0, and it encloses an area equal to half that of the ellipsoid. It is however also obvious that this curve cannot be a geodesic circle, for if it were, the pole of the circle must be the extremity of the longest axis of the ellipsoid, and the lengths of geodesic lines from this point to the ellipse in question are not all equal. We will however examine Minding's solution. Let a series of geodesic lines be drawn on a given surface all starting from a fixed point. Let s denote the length of a portion of one of these measm*ed from the fixed point, yjr the angle which the selected geodesic line makes at starting with some fixed line on the surface; 22 338 MINDING. thus s and -v/r serve as co-ordinates to determine a point on the surface. Now let (f) be sucli a function of s and -v/r that <^c?T/r represents the length of an element of the geodesic circle which passes through the point {s, •xlr) ; then (f) will he a known function because the surface is supposed a given surface. With this notation it will readily follow that the length of the perimeter of any curve is expressed by the integral j'i^Jd^f+'ids)'' between suitable limits; and the area of the enclosed surface is expressed by I ^d-^ ds between suitable limits. Hence by the usual considerations we have to find the minimum of where A is a constant. Minding then proceeds thus. We have for determining the curve of shortest perimeter the equation U {'^f{dff-\-{dsy-\- B jUdf ds = 0. For brevity put dP^= j>^{d->yf+ [dsf, and suppose that only i/r varies since it is known that the two equations which are obtained by varying s and "y^ must coincide ; thus we obtain this gives the following as the differential equation of the curve of least perimeter, This equation will be satisfied by the supposition ds = 0, as it is . easy to see. For it follows from this supposition that dP= (fydyjr, so that the differential equation becomes f ^J d-\lr-d(j> = 0, and this is identically true, whether ^ depends on -x/r or, as in some cases may happen, is independent of ^Jr. MINDING. 339 This is Minding' s process. It appears from tliis process that when we take the variation of the proposed expression, the term remaining under the integral sign is Hence the equation for determining the required curve is Minding in effect multiplies the expression on the left-hand side cls d^ • • of this equation by -j-y , and then puts -jj = as a solution. This is of course unsound. We may put the solution in a slightly different form. Minding really takes s as the independent variable; it is however more natural to take i/r as the independent variable. The double integral jj is a function of s and -v/r this equation connects s and i/r, and shews that s is a function of y{r, so that ds . -j-r IS not zero. ay 22—2 340 GOLDSCHMIDT. 308. Goldsclimidt. Determinatio sujjerficiei minimce rotatione curvce data duo jpuncta jungentis circa datum axem ortce. Auctore Benjamin Goldschmidt. Gottingen, 1831. This essay obtained a prize from the university of Gottingen in 1831 ; it occupies 32 quarto pages. The problem discussed is to find the curve joining two given points which by revolving round a given axis will generate a minimum surface. The problem is solved in three different ways by using different formulse for the area of a surface of revolution, and the result is, as is well known, that the sm'face is in general that obtained by the revolution of a catenary round its base. The author then investigates the pos- sibility of drawing a catenary which shall have a given base and pass through two given points. The conclusion is that sometimes two such catenaries can be drawn, sometimes only one, and some- times no catenary. When no catenary can be drawn it is inferred that the surface consists of two planes formed by the revolution round the axis of the perpendiculars from the given points on the axis J these planes may be supposed connected by means of the portion of the axis which they intercept between them. There is no investigation of the terms gf the second order to shew that a minimum really is obtained. In the course of the essay some interesting properties of the catenary are noticed ; thus on page 17 is given a simple geometrical method of drawing a tangent to a catenary ; on page 18 it is shewn that all the curves formed by varying the parameter c in the equa- X _ X tion 2y = c{^-\-e ') touch two straight lines passing through the origin ; on page 26 is given a simple geometrical method of deter- mining the vertices of the two catenaries which have a given axis and pass through two points equally distant from that axis. A short account of Goldschmidt will be found in the Monthly Notices of the Royal Astronomical Society. Vol. 12, page 84. 309. Poisson. Crelles Mathematical Journal, Vol. 8, pages 361, 362. 1832. This article is entitled Note on the^ surface of which the area hetween given limits is a minimum. We give a translation of it. One of the first applications which Lagrange made of the Cal- poissoN. 341 cuius of Variations was to determine the surface of whicli the area between given limits is a minimum. This was a very favourable example for shewing the advantage of his new calculus over the in- genious methods which had preceded it; for it would have been difficult to extend these methods to the maxima and minima of double integrals, and therefore to questions concerning surfaces. The equation which Lagrange found is, as is well known, a partial differential equation of the second order. Monge integrated it in a finite form, but by considerations which appeared inadmissible, and which gave rise to long discussions between him and Laplace. Legendre afterwards obtained the same integral by a transforma- tion applicable to a large class of equations of the second order, so that no doubt remained as to the con-ectness of the result. (Lacroix, Differential and Integral Calculus^ Vol. 2, page 622.) Unfortunately no advantage could be drawn from this integral, which involved imaginary quantities and was expressed by a system of three equations between two auxiliary variables and the current co-ordinates of the surface. But besides the difficulty which results from this form of the general integral, in which it appears, to say the least, very difficult to determine the arbitrary functions, there is another difficulty arising from the number of these fanctions which the question can admit. In fact the problem of a minimum area comprises two dis- tinct questions ; either two closed curves are given and we require to connect them by a zone of surface of which the area shall be the least possible, or else only one closed curve is given and we have to find a surface such that the area of the portion bounded by this ciu-ve shall be a minimum. When, for example, an aper- ture is made in the surface of a vessel which contains a fluid, the area of the surface by which we must multiply the velocity and the time of the movement in order to calculate the volume of the fluid discharged is precisely the minimum area corresponding to the second case of the problem, which thus presents a useful application. In the first case the question and the complete integral which has been found have the same degree of generality, and the two given cm-ves determine implicitly the two arbitrary functions whicli this integral includes. In the second case, on the contrary, the 342 poissoN. given curve can only serve to determine one arbitrary function. One of these functions will tlien remain undetermined, and the integral will thus have more generality than the question which it serves to solve. If the given curve is plane the surface required is the plane of this curve. If it is a curve of double curvature this surface is not known a priori, but it ought to be some definite single surface, and the problem is not solved so long as there remains anything undetermined in the equation. In order to resolve this difficulty I have considered specially the case in which the required surface does not deviate much from a given plane. By putting the integral of the partial differential equation under a form which difiers from that hitherto used, I have found that the expression of one of the current co-ordinates as a function of the other two contains terms which become infinite at a point of the minimum area, in the second of the two cases of the problem ; and these must be suppressed as foreign to the problem. In the first case these terms retain a finite value through the whole extent of the zone of surface which is to be determined, so that while they are to be suppressed in the other case they are to be retained in this. By this means the expression for the ordinate of any point of the surface has in each case the degree of gene- rality which the question requires. Then, by the method which I have used in other memoirs, all the arbitrary quantities which enter into this expression are determined, by means of the two limiting curves of the minimum zone in the first case, and by means of the single curve which bounds the minimum area in the second case. In this manner the solution of the problem is completely finished in the two parts which it presents, and which form two distinct questions with reference to the determination of the arbi- trary functions, although they depend upon the same differential equation. The Memoir from which this note is extracted will appear in another number of this Journal. [The memoir in question seems never to have been published.] 310. Pagani. CrelUs Matliematical Journal, Vol. 15, pages 84—99, 1836. PAGANI. BJORLING. 343 Tliis article is entitled Solution of a prohlem relating to tlie Calculus of Variations. The problem considered is that which was solved bj Gauss ; see Chapter ill. Before considering the problem Pagani gives a brief investigation of the variation of a multiple integral. He arrives at the formulas contained in Ostrogradsky's Memoir ; see Art. 128. He then gives some remarks on the inte- gration of the expressions when the number of the variables does not exceed three. The Memoir contains nothing that will not be found in Osti'ogradskj, and from its brevity it would be difficult for a student who had not access to other works on the subject. 311. Bjorling. Calculi Variationum Integralium Duplicium Exercitationes. Auctore Em. Gabr. Bjorling. Upsal, 1842. This treatise contains 57 quarto pages. The author refers to the memoirs by Poisson and Ostrogradsky, and expresses his surprise that neither of these mathematicians applied his general formulae to the question of determining the surface of minimum area. Pie proposes to consider this problem. He gives by way of introduction an investigation of the variation of a double inte- gral, with some remarks on the limiting equations which must be satisfied in order that the variation may vanish. This part of the treatise is taken from Ostrogradsky. This introductory part occu- pies the first 19 pages. The author then proceeds to the problem of the surface of minimum area, and he arrives at the well-known result that such a sm'face must be determined from the equation (1 +/) t - 2pqs + (1 + 2') r = 0, where the usual notation is adopted. Before considering this equation generally he gives two special examples in which it is satisfied ; one example is a surface of revolution, and the other a ruled surface. The discussion of these examples occupies pages 20 — 28. Then pages 29 — 50 are devoted to the solution of the general partial differ- ential equation given above. Bjorling quotes Monge's solution; but by means of transforming the variables he obtains the solution under another form, which he considers more suitable than that of Monge when we have to determine the arbitrary functions involved. 344 BJORLING. The author refers for Monge's solution to Monge's Application de V Analyse a la Geometrie, and to Lacroix, Traite du Gale. Diff. et Integ. Vol. 2, page 630. Monge's result is also established in De Morgan's Differential and Integral Calculus, pages 473, 474. The last seven pages of the treatise form an appendix in which the author briefly discusses a particular case of the problem of determining a solid which has a maximum volume while the area of the surface is given. It will be seen from this account of the treatise that it con- tains very little which strictly belongs to the Calculus of Variations ; in fact it should rather be considered as an essay on the integration of the partial differential equation given above. We may observe that the part of the treatise which relates to the integration of the equation is reproduced by the author in an article in Grunert's Archiv der Mathematik und Physih, Vol. 4, pages 290 — 315, 1844. The following four points of interest may be noticed in the treatise. (1) The author before considering the general problem takes the case of a surface of revolution ; he then arrives at the known result that the surface must be that which is formed by the revolution of a catenary round its base. Supposing that the surface is to connect two given circles which have their planes perpendicular to the axis of revolution and their centres on this axis, he obtains equations for determining the constants involved in the equation to the catenary. He then asserts that the surface thus obtained is that which has the minimum area out of all possible surfaces that can be drawn so as to connect the two given circles, and not merely the minimum area out of all surfaces of revolution. He does not explain this remark. Perhaps he means that we are first to conclude that in the case considered the surface must be one of revolution ; suppose, for example, we divide it into two parts by a plane containing the axis, then if the two parts are not symmetrical one of them will generally be of greater area than the other, we can then replace the. part which has the greater area by a part symmetrically equal to the other part, and thus obtain a less total area than that which was assigned as the minimum. BJORLING. 345 Or perhaps the author argues that as the surface of revolution which he has obtained satisfies the general partial differential equation of the problem, and also satisfies the limiting conditions, it must be the surface required. (2) Bjorling discusses another particular example before con- sidering the general equation, namely, among all surfaces which can be formed by the motion of a straight line which always remains parallel to a fixed plane, to determine that of minimum area. Take the plane of {x, y) as that to which the generating line is always to be parallel ; then we have to find a relation between X, y, and z, so that the following partial differential equations may be satisfied, gV — 2j)qs +pH = 0, (1 +/) < - 22;2S + (1 + 2^) r = 0. The result is X— a = {y — h) tan — j — . This result is however more general than appears from Bjorling's treatise. It has been shewn by Catalan that out of all ruled sur- faces the surface determined by the equation just given is the only one which satisfies the condition for a minimum area; see Liou- ville's Mathematical Journal, Vol. 7, pages 203 — 211, 1842. This theorem is also proved by Bonnet in the Journal de TEcole Poty- technique, Cahier 32, page 134, 1848 ; it is there ascribed to Meunier. (3) In the appendix which extends from page 51 to the end, Bjorling considers the following problem; among all surfaces of revolution to find that which has a given area and includes a maximum volume. He obtains the differential equation to the generating curve, and shews that this curve is that which is traced out by the focus of a conic section when the conic section is made to roll on a fixed line. This result he states is due to Delaunay; and he refers to the Journal called L'lnstitut, Number 394, 1841. (4) On page 4 of his treatise Bjorling points out an important misprint in Poisson's Memoir; see Art. 107. 346 BERTSAND. 312. Bertrand. Liouville's Mathematical Journal, Vol. 7, pages 55 — 58, 1842. This article is entitled Note on a 'point- in the Calculus of Variations. Suppose we have to find the maximum or minimum of I TJdx, while Vdx is to remain constant ; then the rule which was given bj Euler is that we must find the maximum or minimum of \{V-\- cU) dx where c is a constant. Bertrand's object is to prove this rule. He says that his proof is not so simple as that which is commonly given, and which involves no calculation ; but the com- mon proof appears to him unsatisfactory, for it only shews that the solutions obtained do satisfy the conditions of the problem, but not that they are the only possible solutions. Suppose then that I Udx is to be a maximum while Vdx J a J a remains constant; then we know that the variation S Udx J a must be zero whenever the variation S I Vdx is zero. Sup- pose for simplicity that the terms outside the integral signs in the ordinary expressions for these variations vanish. Then I cou dx must vanish whenever I cov dx vanishes, where u and v J a J a are certain functions derived in the well-known manner from U and V respectively, and co admits of all values. Now it is obvious that we can satisfy this condition by putting u = cv, where c is a constant ; for then the two integrals have a constant ratio whatever u may be, and therefore they vanish simultaneously. But we wish to prove that this relation u = cv is not only sufficient but necessary. Suppose then that - is not a constant, and let - =f{x) ; then we shall shew that there cannot be a maximum. For we shall shew that it is possible to take co such that / wv dx vanishes while BERTEAND. 347 (ovf{x) dx does not vanisli. For we may suppose that cu is zero J a for all values of x except when x lies between h^ and h^ or between ^3 and h^ ; then we can take a such that (ov dx+ j (ovdx = 0. For we can suppose that h^ — h^ is so small that the sign of v does not change while allies between h^ and h./. and also that h^—Ji^ is so small that the sign of v does not change while x lies between Ag and h^ ; then we can make m have an unchangeable sign during each interval, and choose the same sign or contrary signs for the two intervals according as v has contrary signs or the same sign. By properly choosing h^^, h^, 7i^, andA^ we can ensure that /"(a?) does not change sign while x lies between h^ and h^ or between h^ and h^ , and that the value of f(x) throughout one of these intervals is always greater than throughout the other. Thus /, o}vf{x) dx + I (ovf{x) dx hi J hg will not be zero when the values of co are adopted which we have supposed used to make I (ov dx+ \ 03V dx J hi J A3 zero. Thus there is not a maximum. Therefore there cannot be a relative maximum or minimum unless - is constant. V Bertrand then considers the case in which the terms outside the integral sign in the two original variations do not vanish. It is however unnecessary to notice this part of his article ; for what has been already given shews that there cannot be a solution at all of the problem proposed unless - is constant, and the ordinary method shews, as Bertrand himself admits, that we can get a solu- , . u tion by supposmg - constant. V 348 BEETEAND. 313. Bertrand. Liouville's Mathematical Journal, Vol. 7, pages 212—214, 1842. This article is entitled Note on a Theorem in Mechanics. The following theorem is proved; let there be two curves with their concavities downwards and terminated at the same extremities ; then a particle moving under the action of gravity will take a longer time to describe the upper curve than the lower curve, the initial velocity being supposed the same in the two cases. Take the axis oi y vertically downwards, and the origin so that ^l^gy may be the velocity when the ordinate of the particle is y. Then the time t of describing the arc is determined by the equation ^0 V 2gy ) Now from the usual expression for ht we shall obtain by re- duction J*..y»(l+y')«2V^ Now y" is positive because the concavity of the curve is sup- posed downwards ; and since we pass from the upper curve to the lower by assigning a positive value to hy, it follows that in pass- ing from the upper curve to the lower ht is negative. Thus the time of motion is diminished in passing from one cmwe to another which is infinitesimally lower; and therefore a fortiori the time of motion is diminished in passing from one curve to another which is at a finite distance below the first, provided the passage can be effected through a series of curves indefinitely close to each other aU having their concavities downwards, that is, provided the two extreme curves themselves both have their concavities downwards. Bertrand uses the same method to shew that a convex arc is shorter than another which encloses it ; and he intimates that the same method may be applied to shew that the area of a convex surface is smaller than the area of another which has the same boundary and which encloses the first. 314. Delaunay. Liouville's Mathematical Journal, Vol. 8, pages 241—244, 1843. DELAUNAY. BONNET. 349 This article is entitled Note on the line of given length lohich includes a maximum area on a surface. The area is supposed to be bounded on three sides by curves which project on the plane of {x, y) into straight lines, two of them parallel to the axis of y and the other parallel to the axis of x ; the fourth boundary of the area is supposed to be the curve required, which is to have a given length and to include with the other boundaries a maximum area. The integral to be a maximum is therefore I dx dy \/{l+p^+q^), J a J c where the superior limit in the integration relative to y is the ordinate for any point of the required curve. Moreover the length of the curve is supposed given. Thus the problem coincides with that discussed by Minding and others ; see Art. 307. 315. Bonnet. Journal de VEcole Folytechniqiie. Cahier 32, pages 1—146, 1848. This Memoir is entitled On the general theory of Surfaces. It contains many interesting results with respect to geodesic lines, but it is not very closely connected with our subject ; there are however three points which may be noticed here. (1) On pages 37 — 39 the equation to the geodesic lines on any surface is obtained by means of the Calculus of Variations. (2) On pages 44 — 46 the problem is solved by means of the Calculus of Variations which had been considered by Minding and Delaunay ; see Arts. 307 and 314. (3) On pages 134 — 136 is a note relative to the ruled surface which has at every point its principal radii of curvature equal and of opposite signs. It is stated that Meunier was the first person who proved that the heligo'ide gauche is the only ruled surface which has the property in question. Eeference is made to solutions by Legendre and Olivier ; and it is stated that other solutions have been ' given by writers in Liouville's Journal. Bonnet then gives a geometrical proof of the theorem originally established by Meunier. Bonnet's treatment of the problems (1) and (2) by the Calculus 350 HORNSTEIN. of Variations is very interesting, but it is too closely connected with the notation and results of his Memoir to be extracted. 316. Hornstein. Dissertatio de Maximis et Minimis integra- lium multi^olicium guam pro gradu Doctoratus in celeberrima Uni~ versifate Bonnensi consequendo eldboravit auctor C. Hornstein. Vienna, 1850. This treatise consists of 26 quarto pages. No reference is given to preceding writers, but the treatise is obviously constructed under the guidance of the memoir by Cauchy which we have described in Chapter viii. Hornstein adopts that modification of Cauchy's notation which we have given at the bottom of page 214. The treatise consists essentially of two investigations. (1) An investigation of the variation of a double integral : this is such an investigation as we have given in Arts. 183 and 184. Hornstein gives completely the terms which arise from differential coefficients up to the second order inclusive, and indicates some of the terms which arise from differential coefficients of a higher order. (2) An investigation of the variation of a triple integral ; Hornstein gives completely the terms which arise from differential coefficients up to the second order inclusive. This is similar to the investigation which we have given in Art. 195, so far as the terms arising from differential coefficients up to the first order inclusive. The investigations are given very clearly, and the complicated expressions which necessarily occur have been very accurately printed. 317. Ostrogradsky. Memoir e sur les equations differentielles relatives au prohleme des Iso^perimetres. This Memoir was read to the Academy of Sciences at St Petersburg, on November 29th, 1848, and was published in 1850, in the Memoirs of the Academy. The volume which contains the memoir belongs to the sixth series ; it is the fourtli volume of the department of naathematical and physical sciences, and the sixth volume of the combined departments of mathematical, physical, and natural sciences. The memoir occupies pages 385 — 517 of the volume. OSTEOGRADSKT. 351 Suppose F to be a function of an independent variable t, and of tlie variables x^, x^, ... a?„j, wliicli are supposed to be functions of t, and of the differential coefficients of these functions with respect to t. Moreover suppose that V involves differential coefficients of each fanction x^, x^, ... x„^ up to tliat of the n^^ order inclusive. If I Vdt is to be a maximum or minimum S I Vdt must be zero. Bj the known principles of the Calculus of Variations this leads to 7)1 differential equations each of the order denoted by 2n. Now it is shewn by Ostrogradsky that these differential equations are equivalent to a certain set of 2m7i partial differential equations of the first order. The object of the first part of Ostrogradsky's Memoir is thus the same as that which was afterwards considered by Clebsch in the first part of his second Memoir; see Art. 286. Ostrogradsky then enters at great length into the subject of the integration of the equations which are thus obtained, and the consideration of some remarkable properties connected with the equations. The memoir is rather difficult and not very correctly printed. It is very slightly connected with the Calculus of Variations ; its proper place is among the series of modern researches on the equations of Dynamics, and on the theory of the variation of the arbitrary constants ; to these subjects Ostrogradsky often alludes. The following points of interest may be noticed. In pages 419 — 430 Ostrogradsky makes some observations on that part of the Mecamque Analytiqiie in which Lagrange deduces the equa^ tions of motion in Dynamics from the principle of Least Action combined with the principle of Vis Viva. Ostrogradsky says that Lagrange's analysis is inexact (page 424). The principle on which Ostrogradsky founds his objection is, that by virtue of the equation of Vis Viva there is a relation between certain variations which Lagrange assumes to be independent (page 423). The part of the Mecamque Analytiqiie to which Ostrogradsky refers is that on page 296 and the following pages of the first volume ; in one place Ostrogradsky refers to page 229, which must be a misprint for page 299. 352 SCHELLBACH. In pages 472—480 Ostrogradsky applies his general theory to some examples ; these are of great use as illustrations of his theory. In a note he says that he omits other illustrations because he has found during the printing of his memoir that it was possible to generalise and simplify these applications, and also that the general theory could be simplified and receive some development ; this he promises to shew in a future memoir. On page 512 Ostrogradsky indicates an important application of a formula originally obtained by Poisson, which application Poisson himself appears not to have observed. 318. Schellbach. Crelles Mathematical Journal, Vol. 41, pages 293—363, 1851. This Memoir is entitled Problems of the Calculus of Variations. The author states that students of mathematics often find the Calculus of Variations a difficult subject ; he accordingly considers some problems which are usually treated by the Calculus of Varia- tions and solves them without using the methods of that Calculus. His processes resemble those which were used by the early writers who solved such problems before the Calculus of Variations was reduced to a system. The memoir is interesting and instructive, especially for a student who is examining the foundations of the subject. The memoir consists of 35 sections ; we will indicate briefly the contents of these sections, and then give some specimens of the investigations. (1) The ordinary formulse for solving problems of maxima and minima are quoted from the Difierential Calculus. (2) A curve of given length is to be drawn between two fixed points so as to include with the axis of x and the bounding ordinates a minimum area. The problem is solved by first considering the case of a polygon, forming the necessary equations by (1), and then proceeding to the limit. (3), (4), (5) contain other solutions of the problem in (2) . In (6) the problem, is modified by supposing that the ends instead of being fixed are to lie on given curves. (7) The problem we have solved in Art. 99 after Poisson. (8) To 6CHELLBACH. 353 find the curve which joins two given points and by revolution, round an axis in its plane generates a minimum surface. (9) To find the curve which by revolution round an axis in its plane generates a maximum or minimum volume, the ends of the curve lying on given curves. (10) A general discussion which amounts to finding the usual equation for a maximum or minimum in any integral expression with one dependent variable. (11) To find a curve such that the area between the curve and its evolute may be a minimum. (12) The solid of revolution of least resistance. (13), (14), (15), (16) and (19) The brachistochrone and allied problems. (17) The problem discussed by Minding and others; see Art. 307. Schellbach states that it has been discussed by another mathe- matician besides Minding and Delaunay, but he does not give a precise reference. (18) A curve of given length is drawn on a given surface ; find the curve so that the volume determined by the curve and its orthogonal projection on one of the co-ordinate planes may be a maximum or minimum. (20) A problem which we shall consider presently; see Art. 320. (21) The curve which has its centre of gravity at a maximum depth. (22) The curve which bounds an area having its centre of gravity at a maximum depth. Sections (23) — (29) contain investigations which are not very closely connected with the Calculus of Variations ; we shall recur to them again; see Art. 322. (30) To find a surface having a given boundary and a minimum area. (31) General investigation of the maxima and minima of double integrals. (32) General investi- gation of the maxima and minima of triple integrals. (33) and (34) The problem which Poisson quotes from Euler, and the problem which Poisson himself considers; see Arts. 118 — 120. (35) The transformation of the equations of motion in Dynamics given by Lagrange in the Mecaniqiie Analytique; see De Morgan's Differ^ ential and Integral Calculus, page 520. 319. As an example of Schellbach's solutions we will take the problem of determining the brachistochrone when a particle moves in a resisting medium under the action of gravity ; see section (14) of the memoir. Instead of supposing the particle to describe a curve we will suppose it to describe a polygon of n sides, each side being ulti- 23 354 SCHELLBACH. mately made indefinitely small. Take the axis of x horizontal, and that of y vertically upwards. Let x^ , y^ "be the co-ordinates of the initial point ; x^^y^ the co-ordinates of the beginning of the second side of the polygon; x^^ y^ the co-ordinates of the "begin- ning of the third side of the polygon ; and so on. Let hs^ be the length of the first side of the polygon, v^ the velocity, supposed uniform, with which it is described ; let Zs^ be the length of the second side of the polygon, v^ the velocity, supposed uniform, with which it is described ; and so on. Then the whole time of motion is 85„ Zs, Ss„ . Bs„ -^ + -^ + ^+ + -'m-l ^n-i V, V^ V^ We have then to make this time of motion a minimum. We must first however determine the connexion between the velocity at any point and the co-ordinates of that point, by me- chanical principles. Suppose a particle to be moving on a curve ; let p denote the reaction of the curve, gio the resistance where w is any function of the velocity ; then the equations of motion are d^x dy dx d^y dx dy Eliminate p from these equations ; thus dv^= — 1gdy — 2gio ds. Assume v^ — ^gu, so that du -^^ dy ■\- wds = ^ (1). Now let us return to the supposition that the motion is to take place on a polygon and not on a curve ; then from the equation last written we obtain the following n equations, W.-Wi +2/2-2/1 + ^^1^^1=0 W„ - W„-i + 2/n - y»-l + ^^n-l S^n-l = ^ The expression to be made a minin\um is (2). ^!o+i^ + + ^" n-x V^O 'V'^i '^^'l SCHELLBACH. 35^ which we shall denote by T. Then we may consider T as a function of 2n imknown quantities, namely x^, y^, x^, y^, , ar„_„ y^^, and we must determine the values of these quantities so that T may be a minimum. Now by the ordinary principles of the Differential Calculus we may use the method of indeterminate multipliers in order to take account of the conditions expressed by the equations (2). So that we may consider we have to find the minimum value oiT+tkrM,, where ^^r = Ur^x-u,-\-y.,^^-y,.-^io,hs^, \. is a constant, and the summation indicated by ^ extends from r = to r=w— 1 both inclusive. We shall now differentiate T + ^\M^ with respect to each variable, and equate each differ- ential coefficient to zero. Let us take for example the variables x^ and y/, each of these occm-s in hs^ and in hsr_^ ; for ( Va)' = ix, - x,_^Y+ {y, - y,_^)\ ■ and {Zs,Y = {x,^^ - x,y+ (y,^j - y,y ; moreover y^ occurs explicitly in 31^ and in lf^_j . Thus by differ- entiating with respect to x^ we get 1 Sa7r_i 1 Sa?r ^ ^37r_j Bx^ _ where Bxj._^ is, put for x^ — Xr_^ and 8x^ for a^^+i — x^. The above equation may be written therefore by proceeding to the limit and integrating we obtain -r- -J- + f^w -f- = a (3j, ■ VM as as where a is a constant. Here we have dropped the suffix r — 1, that is, we use w, to, \, -7- , as representing any one of the corresponding quantities with its appropriate suffix. 23—2 356 SCHELLBACH. In the same manner by differentiating with respect to y^ we obtain this equation may be written therefore by proceeding to the limit and integrating we obtain J_^ + xJj + X=i (4), f^u as as where 5 is a constant. Equations (3) and (4) are the differential equations of the problem; they agree with the results obtained by the ordinary methods ; see for example Mr Jellett's treatise, pages 298 — 300. From (3) and (4) eliminate X; then with the help of (1) we shall obtain du x = a y = w (bw + -j-jdu (i-»'){(*«,+J^J-«Mi-»')} r du As w is supposed a given function of u we obtain from these two equations x and y as functions of an auxiliary variable u. 320. In Schellbach's twentieth section the following problem is proposed. The ends of a string of length I are fastened at the points A and B ; the ends of a string of length \ are fastened at the points A' and B'. The four points A, B, A', B' are not supposed to be all in the same plane. A straight line passes from the position AB to the position A'B' so that it moves over the threads I and \ in the same time with uniform velocity, and thus describes SCHELLBACH. 357 a developable surface. Required to determine the forms of the strings so that this surface may he a maximum or a minimum. This problem requires some observations. It is no doubt meant that the straight line is to pass from the position AA' to the position BB', and not, as it is stated above, from the position AB to the position A'B'. The meaning of the problem is best understood by examining the process of solution which the author adopts. Let P, Q denote adjacent points of one of the strings, and P', Q' corresponding adjacent points of the other string. Let a generating line be drawn from P to P' ; let the end at P be supposed fixed, and let the line turn round this end remaining always in contact with P' Q ; thus an indefinitely small conical element is generated. Next let the end of the line at Q be supposed fixed, and let the line turn about this end remaining always in contact with PQ ; thus another indefinitely small conical element is generated. Now it is the sum of all these pairs of elements which the author proposes to make a maximum or minimum. These elements do not form a continuous developable surface in the ordinary meaning of such a term ; for that would require that the following three lines should be in one plane, the line PQ, the tangent to the guiding curve at P, and the tangent to the guiding curve at Q, and there is nothing in Schell- bach's solution to secure this. Moreover there is nothing in the solution corresponding to the condition of moving with uniform velocity over the two curves, which occurs in the statement of the problem; the connexion between the lengths of the two curves described by the moving line in passing from its initial position to any other position is in fact one of the things sought in the solution. Let X, y, z be the co-ordinates of one end of the moving line, s the length of the portion of the string which has been described ; let ^, 7], ^ be the co-ordinates of the other end of the moving line, o- the length of the curve which has been described. Let r denote the distance of {x, y, z) from {^, rj, ^) ; if the first end of the line moves over an. arc ds while the other end remains fixed, the area of the element of surface generated will be ultimately 358 SCHELLBACH. similarly if the second end of the line move over an arc da while the first end remains fixed, the area of the element of surface generated will be ultimately Thus since the lengths of the guiding curves are to he constant, we have, by the usual considerations, to find the maximum or minimum of where m and yu. are constants. Schellbach then expresses r, -7- , -J- ,ds and da in terms of x, y, z, f, >;, ^ and their difierentials ; then by equating the coefficients of the variations to zero in the usual way he obtains equations for determining the required curves. The equations he obtains are susceptible of integration to a certain extent, but the problem cannot be completely solved. Schellbach next considers a modification of the problem ; he supposes that one of the curves is replaced by a straight line of given length and position, and that the other curve is to be deter- mined so as to make the area a maximum or minimum. The solu- tion of the problem in this form can be carried a little further than the solution of the original problem. - 321. In his twenty-first section Schellbach suggests a problem which it will be instructive to examine. '■ ^ (75 is a string of given length which is fastened at A and B; see figure 9 ; A' C'B' is another string of given length fastened at A' and B'-, GDC is another string of given length, the ends of which are constrained to lie on the former strings. Each string is supposed uniform, but the weight of a unit of length is not necessarily the same for all the strings. Required the forms of the curves in order that the centre df gravity of the system may be at a maximum depth. SCHELLBACH. 359 We know from Statics that tlie curves will all be portions of catenaries ; and from Statics we can obtain certain equations for determining the constants involved in the equations to the catenaries. But the point of interest is to deduce these equations hj means of the Calculus of Variations. Take the axis of y vertically downwards, and put j? for ^- . Let w^, w^, w^ he the weights of a unit of length of the strings AGB, A'G'B', CDC respectively. Then we require that the following expression should be a maximum, '^1 \y V(l +/) dx + 10^ h/ \/(l + pl dx + wAy V(l +/) dx, where the three integrals extend respectively over all the elements of the three curves. And the length of each curve is a constant. Hence the following expression is to be added to the former, «i K(l + /) dx + a^ I V(l +/) dx + %W{1 +/) dx ; and the whole made a maximum, a^, a^, a^ being constants, and the three integrals extending over all the elements of the three curves respectively. We then make the variation of the whole vanish. This varia- tion consists as usual of terms under the integral signs and terms outside the integral signs. The terms under the integral signs vanish if we suppose equa- tions to hold of which the type is [loy + a) V(l +i/) = J[i"^y ^ + constant, that is, -^-^^ jr = constant. v(i+y) In this equation w and a are to have the specific value belonging to the specific arc we are considering ; the constant is not necessarily the same throughout, but will generally have five difierent values corresponding to the five arcs, AG, GB, A'G', G'B\ GDG\ - 360 SCHELLBACH. The general relation just obtained shews that each of these five arcs is a portion of a catenary. Now consider the terms outside the integral signs. We adopt the usual supposition that both x and y vary, and we denote by hx^ and hy^ the variations of the point C. Then hx^ and hy^ will occur in three ways, arising from the three curves which meet at G. The complete term involving hy^ will be hy^[L-M-N), where X, M, N are respectively the values at the point C of the expression of which the type is ^^^ ir- j obtained from the curves AC, CB, CD respectively. Thus the equation L-M-N=0 agrees with what we should obtain from the statical principle of equating the sum of the vertical tensions at C of the two upper curves to the vertical tension of the lower ; for the value of ^ found from the curve BC at (7 is negative. Similarly by equating to zero the coefficient of Sx^ we shall obtain an equation coincident with that which we should obtain from the statical principle of equating the sum of the horizontal tensions of the curves BC and. CB at C to the horizontal tension of the curve CA. We have theoretically enough equations to determine the con- stants. For we have five constants from the general relation which we have found above when it is applied to the five arcs, and five more constants would be introduced by integrating that general relation; we have also the three constants a^, a^, a^; thus there are thirteen constants on the whole. Now we have found two equa- tions among the constants from the conditions which subsist at C, and similarly we should obtain two more equations from the con- ditions which.: subsist at C; the four fixed points^, B, A', B\ furnish four equations ; the known lengths of the curves furnish three equations'; the fact that three arps intersect at C fmrnishes one equation, and the fact that three arcs intersect at C furnishes one equation. Thus on the whole we have thirteen equations. SCHELLBACH. 361 322. From page 336 to page 347 of Schellbacli's memoir is occupied with the investigation of some simple maxima and minima problems in mechanics and optics ; this part of the memoir is interesting, though very slightly connected with the Calculus of Variations. The following example may be taken as a specimen of these investigations. Suppose we require the form of a solid of revolution of given mass which shall exert the greatest attraction in a given direction on a given particle, the attraction varying as any inverse power of the distance. Take the given particle as the origin and the given direction as the line from which to measure angular distance ; let r, 6 be the polar co-ordinates of any point in a fixed plane passing through the given direction. Then if the attraction vary as the ?i*'* power of the distance the attraction of an element whose co-ordinates are r and 6 may be denoted by /*?•" ; and the resolved part of this at- traction in the given direction will be yu.r" cos 6. Hence the equation yu,?-'^ cos 6 = constant represents a curve such that a given element placed at any point of it will exert the same attraction on the given particle. Hence this equation represents the curve which by revolving round the fixed direction will generate the required solid of maximum attraction, the constant being determined so as to give to the solid the prescribed volume. It is obvious that such is the case because the surface we thus obtain separates space into two parts, and any particle outside the surface exercises a less attraction than it would if placed within the surface, n being supposed negative. The result we thus obtain may of course also be obtained by tlie ordinary methods of the Calculus of Variations. 323. In pages 357 — 360 of the memoir we have an interesting application of the Calculus of Variations which Schellbach states that he has taken from a memoir by Jacobi in the 36th volume of Crelle's Mathematical Journal. We will explain this application rather more fully than Schellbach does. Suppose in the triple integral \\\Gdxdydz that 6^ is a function « dv dv dv ■, • r x- /« AT oix,y,z,v,-j-, -J- , -J- ; where v is a lunction oi x, y, z. JNow 362 SCHELLBACH. suppose that x, y, z are expressed in terms of new variables \. ii, v by means of tlie equations X =/ (\, fj.,v), y =f^ (X, fJ^^v), z =/3 (X, ^l, v) ; then V becomes a function of X, /*, v, which we shall denote by ^, and a becomes a function o^\, /jl, v, -^, -^ , -£, which we shall denote by F. Then by the known theory of the transformation of multiple integrals we obtain jjtGdxdydz = l[lTUd\dfxdv (1), where 11 is a known expression which involves the differential coefficients of x, y, z with respect to \, /j,, v, so that 11 is in fact a known function of X, fi, v. Now take the variations of both mem- bers of this equation ; these variations will be equal, and the unin- tegrated portions will be separately equal. Thus we obtain the result QSvdxdy dz= j I iRB(f>d\ dfi dv, where Q is an expression derived in the well known way from G, and B is similarly derived from m. Now change the variables in the integral on the left-hand side of the equation ; thus 1 1 1 QU Bvd\d/j,dv= \\\EB

is) h' ''° '' '^^ ^f"^"- 364 SeHELLBACH. Then from the general formula (2) after division by V siny* we obtain d\ d\ d^v ~M^'d^^~d£' 1 [ d [^^ . dv\ d f ^sin/^j- + / . dv\ d_( J}^ dv\\ \ djx) dv\&va. fjb dvj) X^ sin fi [d\ \ dxj dfj, \ _ 1 d f^ dv\ 1 d f . dv\ 1 d^v V d\ \ d\J \^ sin fi d/j,\ d/Mj V sin^/i du' 1 d"" (\v) 1 d f . dv\ 1 d\ X dX X' sin fi dfj, \ dfj,/ X^s'io^ fidv^' Thus we obtain the well-known transformation first given by Laplace. d^z d^z As another example let it be proposed to transform -7-^ + -7-2 into an expression involving r and ^, where x = r cos Q and y = r sin 0. T-r dz dz dx dz dy ^ dz . _ dz Here -T- = j--i- + ^--/- = cos^^-+sm^-^, ar ax dr ay dr dx dy dz _dz doa dz dy _ . ^dz ^ dz dd dx dd dy dd dx dy ' (dz-^ (dz^ (dz^ 1 (dz^ ^-A-^Yrdrdd. Therefore as in (1) Then as in (2) we obtain d^ d'^z _\{d_f dz\ d n dz\] ^^,1^,1^ d^^'^df" r\dr V drj'^de [r dd)} ~ dr' '^ r dr r' dO"' Thus, as Schellbach observes, the transformation used by Poisson can be readily effected. See Art. 119. SPITZER. 365 324. Spitzer. Grunert's ArcMv der Mathematik und Phi/sik, Vol. 23, pages 125, 126. 1854. This article is entitled Note on the shortest lines 07i curved surfaces. When a curved surface can be divided by a plane into two symmetrical portions the intersection of the plane and surface, when an intersection exists, is in general a line of minimum length on the surface. The proof is very simple. Suppose in fact that the equation to such a surface, which is divided symmetrically by the plane of xz, is For a minimum line on the surface we must have the integral s=ls/{dx^^df-^d£') a minimum. Put then dz = pdx + qdi/, so that s = hldx"" + dy" + {;pdx + qdyf] then the condition for a maximum or minimum is dy L dy this gives = 0; ^v^Mt^y% \dy_J_dy_ y'+{p+qy')q = 0; V{i +y^+ (i> +9[yr] LV{i + 3/"+ {p + ay'YU this may be reduced to {y"-\■q[v + qy')']^/[l+y•'^■[T'rqyy] = [y' + ip + ^y'h] [V{i + y' + {f + ^yTU'- This equation is satisfied when y = ; for if y = 0, so are also y = 0, y = 0, and ^ = 0. 366 HEINE. As a spHere is divided symmetrically hj any plane which passes through its centre, any great circle of a sphere is a line of maximum or minimum length. 325. Heine. Crelle's Mathematical Journal, Vol. 54, page 388. 1857. This article is entitled Lagrange s Theorem. It consists of a proof of Lagrange's Theorem hy the method previously used by the author for establishing Jacobi's Tlieorems ; see Articles 296, 297. Let y-1if[y)^x (i)r let t/t (a;) =0 denote any function of a?, and denote the differential coefficients of s with respect to « by s', «", Then t {x-hf[x^;\^z-\z'f{x) +^z" [MY- If then ^ [x] be any function of x whatever, f^<^[x)f[x-hf{x)]dx = fj{x)\^z-\zj{x) + ^^z'[f{^^^^^ (2). Put y for X on the left-hand side of (2) ; then it becomes •b J a and therefore by (1), and therefore b (2/)^^^^» where a — a — hf{a), (3=^1) — hf{h). If h be small enough, at least a portion of the interval between a and h will coincide with a portion of the interval between a and /3. Let z be so varied that within this . common interval hz may have GIESEL. LbFFLEE. 367? anj arbitrary value and be zero beyond it. The variation of the right-hand member of (2) will consist of terms free from the integral sign together with I B ^z dx, J a where ff_^M , h d[{x)f{x)] F d^4>{x)[f{x)Y} . And since we must have within the common interval jBBzdx = j8z-£{i/)dx, therefore ~ j>{y) = B. This is in effect Lagrange's Theorem. 326. Giesel. Geschichte der Variationsrechnung. Einladung- schrift zu der Feier des Schroderschen Stifts-Actus im Gymnasium zu Torgau am 5 April, 1857. Torgau, 1857. This is the first part of a History of the Calculus of Variations. It occupies 45 quarto pages, and details the history of the sub- ject from its origin until the publication of Lagrange's memoir in the Miscellanea Taurinensia in 1762. It is a valuable work, and contains numerous quotations and exact references to the original sources. It resembles in some degree the well-known work of Woodhouse, but it is less didactic and more purely historical. There is a brief notice of this treatise by Schlomilch in the ZeitscJirift filr Mathematih und Physih, 1857, Liter aturzeitung^ page 60. Schlomilch commends the treatise highly. 327. Loffler. On the Method of finding the greatest and least values of undetermined integral expressions. This article is printed in the 34th volume of the Sitzungslerichte of the Academy of Sciences at Vienna, 1859. It occupies 30 octavo pages. The article consists principally of remarks on the brachi- 368 LINDELOEF. stoclirone problem ; the remarks appear of no value, but seem to indicate that the writer has imperfectly grasped the subject. 328. Lindeloef. New demonstration of a fundamental theorem of the Calculus of Variatiojis. This article was published in the Comptes Bendus...de VAcademie des Sciences. Vol. 50, 1860, pages 85—88. The fundamental theorem referred to is one given by Ostro- gradsky, which was also proved by Cauchy; see Arts. 127 and 190. The object of Lindeloef 's article is to establish this theorem by the method used by Poisson, in the case of two independent variables : see Arts. 102 and 103. We will give a translation of Lindeloef 's article. It is known that the variation of an integral can be presented under two forms, according as we do or do not vary the inde- pendent variables as well as the unknown functions. We propose to establish the first form, using only the principles of the differ- ential calculus. We adopt the method which Euler introduced into the calculus of variations, and thus regard every unknown function as involving an arbitrary parameter i, and the variation of the function means its differential coefficient with respect to this parameter. We re- gard the variables x, y, z, ... t, as unknown functions of other inde- pendent variables |, t], ^, ...t, and of the parameter i, and the variations of £c, y, z, ... t, mean their differential coefficients with respect to the parameter i. Thus any unknown function u of the variables x, y, z, ... t, -is susceptible of two kinds of variations, since it depends upon the parameter i directly as well as by means of these variables, and it is advisable to distinguish them by a difference of notation. The partial differential coefficient of u with respect to i we shall call the proper variation of u, and we shall denote it by Bii ; the total dif- ferential coefficient of u with respect to i we shall call the total variation of u, and we shall denote it by Du. This distinction does not exist with respect to the va^ables x, y, z, ... t, and we might use either symbol D ov h for their variations; we shall adopt the latter symbol. LINDELOEF. 369 We now propose to investigate the variation of a multiple integral 8=\\\...vdxdydz... when the limits are variable but continuous. For this purpose we first replace the variables x, y, z, ... ^ bj others ^, tji ^^ "• > which we suppose connected with the first hj the difierential equations dx = a^d^ + a^df] + a^d^+ ... + a^dr, ■ dy — 5,f?f + h^dri + l^d^ + . . . + Kdr, dz = CjV 24 370 LINDELOEF. Under this condition tlie limits of the new integral S' will not change with i; its total variation will therefore be J)S=jjj... {TDV+ VDT)d^d7]d^... , and it only remains to develop DT. We observe then that the partial differential coefficients a, b, c, ..., with which the determi- nant T is made, must be of the nature of the preliminary functions X, y, z, ... , from which they spring, and they must consequently be regarded as functions of ^, 97, ^, . . . , varying with the parameter ^. We have therefore immediately da do dc dli dH . In order to determine the sum X t- 3a, we introduce n auxiliary quantities a^, Wa, 0(3, ... a„, by the equations ^l«l + '^2°^2 + ^3^3 + • • • + O'n^n = 1> ^1«1 + ^2«2 + h^S +••'+ K'^n = 0, qwi + CgSa + Cgag + . . . + c„a„ = 0, K'^l +K'^2 +h^3 + • • • + K^n = 0. Solve these with respect to a^ ; thus we obtain a,T = X(±V3...7g, which shews that the product ot^T does not involve any of the quan- tities a. We can prove the same thing with respect to the products From the identical equation LINDELOEF. 371 we obtain immediately On the other hand, if we regard Sa?, Sy, 85;, ..., as immediate fmictions of £c, 3/, ^, ... , we shall evidently have 5. dSx , dSx dSx , «?Sa3 -> d8x -J dSx dSx , dSx K = %-^+*,-^ + o.^ + ...+A,^, 5, £?Sa; , dBx dSx , c?Sa; By means of these developments the sum dT ^ dT ^ , dl ^ dT ^ da^ ^ da^ ^ da^ * <^„ reduces to X^- Ba = -7- T. da ax A similar process would give 2 — Bh=^T dh dy "* and so on. Hence finally \dx dy dz '" dt) ' If we put this value in the expression i?>S'and restore the original 24—2 372 LINDELOEF. variables we have definitely for the variation of the proposed integral This formula is due to M. Ostrogi-adsky, who established it hy the infinitesimal method ; afterwards M. Cauchy arrived at it by other considerations. As to the demonstration just proposed, which depends essentially on a change of variables, it is right to remark that the same expedient had been already employed by Poisson, when he investigated the variation of a double integral. [The part of the preceding article in which it is inferred that the limits of yS" do not change with / seems difficult; and it might with advantage be replaced by the method of Poisson given in Art. 86.] CHAPTEE XIII. SYSTEMATIC TREATISES. 329. We shaU now give an account of the works which have been published as systematic treatises on the whole subject. There are three which from their extent and importance demand particular notice ; these we shall describe in the present chapter. In the next chapter we shall take the remaining works. In each chapter we shall follow the chronological order. 330. The first of the three treatises is that by Dr Gr. W. Strauch. Its title is Theorie und Anwendung des sogenannten VariationscalcuV s. Zurich, 1849. This work consists of two closely printed volumes of large octavo size. The first volume contains 499 pages, and the second 788 ; the first volume also contains a preface of 32 pages. The Preface begins with a sketch of the history of the subject from the earliest period until the publication of Lagrange's Theorie des fonctions analytiques in 1797 ; this sketch is furnished with references to the original memoirs. The remainder of the preface is devoted to an account of the contents of the work and an indication of the points in which the author believes that he has improved or corrected the methods of his predecessors. Of the writers in the present century, Strauch mentions Lacroix, Gergonne, Dirksen, Poisson, Ohm, and Ostrogradsky. It is remarkable however that he takes no notice of Jacobi's theorems, nor does he refer to the memoirs of Gauss, Delaunay, Sarrus, and Cauchy, which we have described in preceding chapters. 374 SYSTEMATIC TREATISES. 331. The work may "be divided into four parts. The first part occupies pages 1 — 356 of the first volume; these pages con- tain all the ordinary theoretical investigations of the suhject, exclusive of those which refer to double or multiple integrals. The second part occupies the remainder of the first volume, and consists of the solution of 60 problems of maxima and minima, in which neither integrals nor difierential coefiicients appear ; these problems are in fact almost entirely examples of the ^ordinary theory of maxima and minima values which is given in the Dififerential Calculus. The third part occupies pages 1 — 211 of the second volume, and consists of the solution of 93 problems of the maxima and minima values of expressions which involve dif- ferential coefficients but not integrals ; these problems thus re- semble that which we have given in Art. 3, from Lagrange. The fourth part occupies pages 212 — 739 of the second volume, and consists of the solution of 135 problems respecting the maxima and minima values of expressions which involve single or double integrals. The remainder of the second volume forms a supple- ment which is chiefly devoted to the theory of relative maxima and minima values. 332. Strauch may be considered as the successor of Ohm, whose methods he chiefly follows. The most valuable part of his work is that which we have called the fourth part. The problems there given are discussed with great fulness and clear- ness, and the terms of the second order are almost always com- pletely exhibited in order to discriminate between maxima and minima values. Strauch is however content with Legendre's treat- ment of the terms of the second order, that is, he generally as- sumes that certain differential equations can be solved, and that the solutions of such differential equations will not introduce quan- tities that can become infinite ; see Art. 5. In a few simple cases however Strauch actually solves the equations which are analogous to equation (2) of Art. 5. With the single exception of the general problem of the shortest line on any surface, all the great problems of the Calculus of Variations occur in Strauch' s collec- tion; and although he has not givto the shortest line on any surface, he has given cases of the shortest line on specific sm'faces. The problems are always accompanied by excellent historical STEAUCH. 375 accounts of their origin and progress. On the whole, although the work contains much that is superfluous, and much that is of very inferior interest, and very little so far as the theory- is concerned which had not appeared before, yet the large collec- tion of carefully solved examples which it contains, recommends it to the notice of every student of the Calculus of Variations. The work is distinguished for remarkable accuracy, both in the investigations and in the printing. 333. We proceed to give a more detailed account of the contents of these volumes. We begin with the first volume. The first section occupies pages 1 — 8 ; it is entitled Propositions which helong to the Differential Calculus. These pages contain nothing of importance ; they are principally explanatory of the notation which the author adopts for distinguishing differential coefficients formed on different suppositions, such as partial and complete differential coefficients. The second section occupies pages 8 — 13; it is entitled Propositions which belong to the Integral Calculus. These pages contain nothing new ; they principally refer to the differentiation of an integral with respect to any parameter which may occur in the expression to be integrated. The third section occupies pages 13 — 20; it consists of an investi- gation of the conditions under which certain homogeneous func- tions will retain an invariable sign. For example, consider the expression Af + 2Bpq^ + C/ + Wpr + 'iEgr + Fr^, • and suppose that A, B, C, D, E, F are fixed quantities, and that ]J, q, r are variables which may have any value ; then Strauch investigates the conditions that must hold among A, B, C, D, E, F in order that the expression may be of invariable sign. The fourth section occupies pages 20 — 69 ; it treats of the development of a function in powers of a variable, the function being connected with the variable by means of an unsolved equation. Thus, for example, on pa^e 44, Strauch proposes to express w in a series of ascending powers of v, the equation which connects u and v being u^ — Snuv + v^= 0. 376 SYSTEMATIC TREATISES. From this equation three series are deduced for u. For it is shewn that we may have V. = A^v" + A^v' + A^v^ + ^y + . . . where A^, A^, A^, A^,... are successively determined, and each has only one value. Or we may have where B^, B^, B^, B^,... 2i>re successively determined. In this case it is found that B^ may have two different values, and as the subsequent coeflacients B^, B^, B^,... are found in terms of B^^, each value of B^ gives rise to a series for u, so that we have two different series for u. Thus, on the whole we have three forms for the expansion of u in terms of v ; this might of course have been anticipated from the fact that the original equation is of the third degree in u, and therefore furnishes three values of u. The method used by the author throughout this section is that of indeterminate coefficients. He is very careful in his ex- amples to obtain all the different expansions which his expressions will furnish; and he has thus given a more complete exemplifi- cation of the method of indeterminate coefficients than is usual in works on algebra. The subject however is not very closely connected with the Calculus of Variations ; and the chief use of this section is in relation to the first series of examples in the book, which as we have said belong to the ordinary theory of maxima and minima. Thus the first four sections of the work are only introductory to the Calculus of Variations. 334. The fifth' section occupies pages 69 — 131 ; it is entitled Theory of the so-called Calculus of Variations. In this section Strauch explains what is meant by a variation, and shews how to find the variations of different expressions. He objects to the word variation as not sufficiently distinctive, since the notion of variables runs through the whole of the Differential Calculus; STRAUCH. 37.7 moreover the word variation is used in a peculiar sense in algebra. Accordingly lie adopts the word mutation instead of variation. We shall however generally retain the usual word. His definition of a variation coincides in fact with that which has been adopted by Euler, Lagrange and Ohm ; see Arts. 22, 15, 55. Let y stand for ^ [x) ; then he supposes j> (x) changed into (f) {x, k), and {x, k) expanded in powers of k by Maclaurin's Theorem ; this expansion he denotes by Then supposing k indefinitely small the series is called the variation of y. The quantity k is considered indepen- dent of x ; thus the variation of -4- is ax "^ dx '^ 1.2 dx "^[3 dx "^ '" ' and by differentiating this series with respect to x we obtain the variation of -y^. , and so on. dx^' Strauch lays great stress on the view he takes of a variation, and asserts that the common method of denoting the variation of y by a single term as By or KSy leads to a number of absurdities and con- tradictions. This assertion seems however quite arbitrary, and a very careful examination of his theory and problems has not afforded any confirmation of it. On the contrary, his method leads to a great and needless complexity in the exhibition of the terms of the second order in the variations of expressions, with the view of discriminating between maxima and minima. The student of Strauch's work will effect a great simplification, without any loss, by supposing that such expressions as S'^y, B^y, ..., are all zero, so as to reduce each variation to its first term. 378 SYSTEMATIC TREATISES. In the latter part of his fifth section Strauch explains a dis- tinction on which he also lays great stress. A function is said to experience a mixed mutation when some of the variables undergo that kind of change which the Calculus of Variations contemplates, and others that kind of change which the Differential Calculus contemplates. We can illustrate this by considering an Integral, although Strauch himself does not introduce Integrals until the next section of his work. In varying an integral then he does not ascribe any variation to the independent variable, but makes changes in the limits of the integration. Thus he calls the whole change in the integral a mixed mutation, since it partly consists of an ordinary change of value of the limits of the independent variable, and partly of a change oi form of the dependent variable which is strictly a variation or mutation. This restriction of variations to the dependent variables seems to possess all the ad- vantages which Strauch claims for it. The sixth section occupies pages 132 — 165 ; it treats of some special points in the theory of variations. This section presents nothing remarkable or important ; it is chiefly occupied with in- ferences which follow from the view the author takes of a variation as consisting of an infinite series of terms. 335. The seventh section occupies pages 165 — 356 ; this con- tains the general theory of maxima and minima values. This section is divided into three parts ; in the first Strauch considers expressions which involve neither integrals nor differential co- efiicients, in the second expressions which involve differential coefficients, in the third expressions which involve also single integrals. Some general remarks and definitions are given on pages X65 — 171, and then the theory of the maxima and minima values of functions involving neither integrals nor differential coefficients occupies pages 171 — 231. This part of the book contains the ordi- nary theory of maxima and minima values which is explained in treatises on the Differential Calculus; the language is rather different from that which is usually employed and the various cases which occur are treated separately with great care ; nothing STRAUCH. 379 however is given whicli miglit not be obtained from the ordinary- treatises on the Differential Calculus. This part of the work is illustrated bj a series of sixty problems occupying pages 357 — 480 of the first volume. These sixty problems are all, with the exception of the last six, ordinary problems of maxima and minima; the last six are of a slightly different character. Take for example problem 55. Let a be a given quantity, and let it be required to determine the function (f) so that the following ex- pression shall be a maximum or a minimum, without assigning a specific value to x, a" - {0 {x)Y + 20 {x) cf) (a) - {<^ {a)Y -2x<\> {x) -2a STEAUCH. 383 and -7-4 ; Iiere lie does not supply an investigation of the dx dy dy terms of the second order. This investigation on pages 713 — 717 does in fact sum up all that Strauch accomplishes with the varia- tion of multiple integrals ; his result coincides with that which we have abeady given after Sarrus; see Arts. 183, 184. Strauch, as we have already stated, does not refer to some of the writers whose works had preceded his own ; see Art. 330. He is consequently disposed to claim as new investigations which had already been made. Thus on his page 574 he supposes that he is the first to investigate the terms of the second order in a cer- tain double integral ; Brunacci however had preceded him ; see Art. 213. Again, on his pages 737, 738 he institutes a comparison between his own results and those of Poisson and Ostrogradsky ; and he justly states that his own are in sonae points more gene- ral. But, as we have stated above, Sarrus had preceded him in the investigation which really involves all that he accomplishes. See Art. 138. We will now consider some special points suggested by the work of Strauch. 337. We have spoken above of the extreme accuracy of the work in general ; we will here indicate a few points which appear tobe incorrect. On page 438 of Yol. ii. a case of the brachistochrone is discussed ; a heavy particle is supposed to be constrained to move on a fixed plane, and there is a resistance which varies as the square of the velocity. Here Strauch obtains the result that the curve becomes a straight line. But he has interchanged the values of the quantities which he obtains from his equations xxiv. and xxxi. ; and he does not observe that the true values render his F{y) infinite and vitiate his solution. He does not observe also that we can resolve the force of gravity into two components, one in the fixed plane and the other perpendicular to it, and then neglecting the latter component the problem is the same as if the particle moved in a vertical plane. The latter remark applies again on page 454. 384 SYSTEMATIC TREATISES. On page 445 of Vol. Ii. Strauch is discussing a problem given by Euler ; the curve is required down which a heavy particle must move so as to acquire the greatest velocity, supposing a resistance varying as the square of the velocity. Strauch exhibits some investigations for discriminating between a maximum and a minimum. His equa- tion XXV. cannot however be allowed, because his equation xxiv. from which he deduces it, is true for the curve which the particle is supposed actually to describe, but not true necessarily for any other curve. On pages 461 and 462 of Yol. Ii. he attempts to shew that it is possible that ., ^ g, can always be equal to a constant B, and yet L vanish when x = a. This is impossible, for .. -^ — ^ is always finite, and .-, ^ ... is less than L. In fact his equation xxviil. shews that y is impossible when ic = a if ^ is not zero. The only conclusion is that B= 0; then ^ = 0, and the curve becomes a straight line, as might have been anticipated. Similar remarks apply to page 466 ; it is impossible that La = and . ,, ^ g. = a constant, unless that constant is zero. 338. Suppose we require the maximum or minimum value of an expression j(f>dx, where ^ involves x, y, and the differen- tial coefficients of y with respect to x. Now the well-known process is to obtain iB(j> dx, and to reduce this expression as much as possible by integration by parts until it takes the form L+\MBydx, where M contains no variation; then we put J/= 0. Strauch has a very singular notion on this subject. He says it cannot be proved that we must have M=0; although he allows that we do get solutions of our problem' thus. Accordingly he pro- poses to try if solutions cannot be obtained by putting S^ = 0; See his Preface, pages xxv. and xxvi. Thus he frequently tries two STEAUCH. 385 processes of solution of his problems. It may be safely asserted that the ordinary view of the necessity of the equation Jf = is sound ; supposing that whatever series of values hy can assume it can also assume the corresponding series of values numerically equal but of opposite sign, without changing the limiting values of the variations. And on examination it will be found that nothing is gained in any part of Strauch's work by paying attention to what he considers a second solution of some of his problems. Let us take for example the case which he himself brings forward in the preface. Required the maximum or minimum value of the expression I {y^ — 2xy + 2^ —p^) dx, J a The ordinary method furnishes the equation A dx that is, J'-^+S^o «• And we have also the limiting equation (l-i>)„8y„-(l-^)„Sy„ = (2). Strauch then proposes the following as another solution. The variation of the proposed expression is ["2 [{y-x)Zy + 2{l -jp) Sp} dx. J a Without effecting any reduction by integration by parts, make this expression vanish ; this we can do by supposing y-x = and 1-^ = (3). The second of equations (3) is in this case consistent with the first, so that we do get a solution. This however is not a new solution; it is comprised in (1) ; for ^ = a; is a particular solution of the differential equation (1); and y = x also satisfies (2). Thus Strauch's supposed second solution is really included, as it should be, in the ordinary solution. , 25 386 SYSTEMATIC TREATISES. Before leaving this part of our subject we will offer some re- marks with the view of guarding against a possible misconception of the principle of equating separately to zero the two parts of the variation of an integral, in order to obtain a maximum or minimum value of the integral. Consider the problem of finding the curve which with its evolute includes a minimum area. Let p denote the radius of curvature at any point of the curve, s the length of the arc of the curve measured from any fixed origin up to this point ; then we require that 1 p ds should be a minimum. Let p receive the variation 3p, and let s^ and s^ receive the increments ds^ and ds^ respectively ; then the change in the integral is I Bpds + p^ ds^ — Pq ds^ •J Sn Here the coefficient of Sp under the integral sign is unity, which cannot be made to vanish ; so that it might perhaps be supposed at once that the solution of the problem is impossible. But the fact is that we cannot prove in such a case that in order to obtain a solution we must make the integrated part and the unintegrated part separately vanish. For when we take the arc s as the independent variable, and pass from one curve to an adjacent curve the length of the arc will in general be changed ; and if we make any change in that part of the variation of an integral which remains under the integral sign, the part outside the integral sign also undergoes a change. In other words, the two parts which constitute the whole variation of a proposed integral are not mdejjendent, so that we are not compelled to make them separately vanish in order that the whole variation may vanish. If we can make them separately vanish we obtain a solu- tion of the problem, subject of course to an examination of the terms of the second order ; but we are not certain that this is the only solution. And if we cannot make them separately vanish we must not therefore conclude that the problem is impossible. The point we are now considering ig perhaps sufficiently obvious ; but as it is sometimes a som-ce of difficulty to students it may be useful to refer to two other examples. M STRAUCH. 387 Suppose we require a curve which has the property of making c + -]ds a, minimum, the ends of the curve being supposed fixed ; c is a constant and r is the radius vector drawn from a fixed pole. The problem is thus equivalent to the following ; assuming the principle of least action in Dynamics, and the ordinary law of attraction, determine the curve which a particle will describe. The result ought to be a conic section, and we shall obtain this result if we adopt the usual independent variable 0, and put a/^'+(i: d6 for ds. But no result wiU be obtained by attempting to determine r as a function of s and operating in the usual way immediately on IVH. ds. Again, suppose we require to describe on a given chord a curve of given length, such that the area included by the curve and the chord may be a maximum. This can be easily solved in the usual way by taking x as the independent variable ; the result is that the curve must be a circular arc. But suppose we take s as the in- dependent variable, and take a fixed point as pole. Then the polar area between the curve and the extreme radii will be s/V'-lsJ*' and as the triangle included by the given chord and the extreme radii is itself constant, we have to make the above area a maximum ; also the length of the given curve is to be constant. Thus in the usual way we have to make the following expression a maximum, /iVi-(sy+4*' where c is a constant. Proceeding in the usual way we shall have the equation 25—2 388 SYSTEMATIC TREATISES. aA^ — r , , + c = — . ^ ' i + constant, ^/-(|y therefore , — — = a, where a is some constant. Therefore 0" = ^ , that is, 1 + / (f) = ~ ; therefore (^-1 =—2 2* drj a^ — r^ From this we should ohtain for the required curve a circle passing through the arbitrary pole ; and this is inadmissible, be- cause the circle is determined by the fact that it is to pass through the ends of the given chord and that its arc cut off by the chord is to have a constant length, so that it cannot in addition be made to pass through an arbitrary point. If 2> denote the perpendicular from the pole on the tangent to the curve, the problem amounts to requiring that Up + c) ds shall be a maximum ; and in this form we see at once that no solution oan be obtained by the ordinary method if we keep s as the inde- pendent variable and endeavour to determine ^ as a function of s. We have hitherto spoken only for simplicity of the use of the arc s as an independent variable ; but our remarks apply also to the use of the arc s as a dependent variable. Thus, taking the example already used, we have but if we adopt the right-hand form and thus treat r as the inde- pendent variable we shall arrive at the same untenable solution as before. The objection to the process is easily seen. Suppose we draw one curve through two fixed points, and then draw an adjacent STBAUCH. 389 curve by changing every s into s + 8s, and also pass from the first curve to a third curve by changing every s into s — Bs; then if we make the second cm*ve to have the fixed initial and final points, the first and the third curves will not in general have the same final points. That is, we cannot change the sign of Bs arbitrarily, and therefore we have no right to conclude that the coefficient of Bs in the part remaining under the integral sign in the variation of the integral must be zero. We may add that the fact that when we use the ordinary variables x and y we must equate to zero the coefficient of the variation under the integral sign, seems more obvious when we ascribe a variation to the dependent variable only than when we also vary the independent variable ; this is an additional argument in favour of an opinion already expressed. See Art. 204. 339. Problems of maxima and minima which involve the product or quotient of integrals are sometimes incompletely solved. Strauch has given some examples for the purpose of drawing at- tention to the point which is liable to be overlooked; see his Preface, pages xxx, and xxxi. This deserves to be illustrated fully, and we will accordingly give two problems in addition to his. I. Determine the form of a curve symmetrical with respect to its axis such that when suspended by its vertex the time of a small oscillation of the segment cut off by the ordinate which corresponds to a given abscissa may be a minimum. Take the vertex as the origin, the tangent at the vertex as the axis of y and the axis of a? vertically downwards; let c denote the given abscissa. The area cut off by the ordinate which corresponds to c is supposed to oscillate about an axis through the origin per- pendicular to the plane of the curve. Then by the principles of mechanics the length of the equivalent simple pendulum is I yxdx Jo and this expression must therefore be a minimum. 390 SYSTEMATIC TEEATISES. Denote the numerator and denominator of this fraction by u and V respectively. Then that - may he a minimum we must have therefore that is, 8w wSv_ Zu — Sv = : V I (y + x^) Sydx / xSydx = 0. Now let - be denoted by I; then I is a constant for our purpose, so that the last equation may be written / (y + x'- Ix) lydx = 0. J n Hencfe in the usual way we infer that y + £c'' — ?a; = 0, and so we apparently obtain a circle as a solution of the proposed problem. The solution however is not yet completed; for we require that - should be equal to I. Substitute for y its value in terms of x which has just been obtained ; then we require that \ \\lx + 2a?) ^/{tx - x'') dx therefore I X \/(Jx — of) dx Jo 2 f - x' \/{lx — x^) dx ilo . I X sjilx — x^) dx J a = ?, 21 3' STRAUCH. .391 therefore that is. ■'0 ^[\x-x^fdx = ^. This is impossible ; so that the proposed prohlem does not admit of a solution. In fact in this problem as there is no limitation about the area we can suppose it to diminish down to an indefinitely small area in the neighbourhood of the origin, and so make the time of a small oscillation indefinitely small. In such a problem as the above, the investigation as to whether such a condition as that denoted hj - = l can be satisfied, is some- times omitted ; in the present case it appears that this condition cannot be satisfied. We will now give a problem of the same kind which does admit of a solution. II. A given volume of a given substance is to be formed into a solid of revolution, such that the time of a small oscilla- tion about a horizontal axis perpendicular to the axis of the figure may be a minimum ; determine the form of the solid. Take the axis of x coincident with the axis of figure, and the axis of 1/ coincident with the line about which the body is to revolve ; let x^ be the abscissa of the lowest point of the body. We have to find the equation to the curve, which by revolution round the axis of x will generate the required solid ; we suppose the curve to lie in the plane of {x, y). By the principles of mechanics the length of the equivalent simple pendulum is I y^xdx •' this expression must therefore be a minimum, while tt I ifdx is 392 SYSTEMATIC TREATISES. to be equal to a constant, namely to the given volume. Hence, by tbe usual principle we must bave y'xdx '' J a minimum, where /3 is some constant. "We will at first vary y, and afterwards examine the terms which arise from a change in the limit x^ of the integrations. Let u and V denote the numerator and denominator respectively of the fraction which occurs in the above expression ; then in order that the expression may be a minimum, we must have Bu uBv , osP» 2^ A r + pSj y ax = 0, therefore Bu~Bv + ^vSr'fdx = 0, that is, I (y* + Si/a;") Bi/dx 2yxBydx + ^v \ 2yBy dx^O. Now let - be denoted by I, and ^v by yQ' ; then I and /8' are constants for our purpose, so that the last equation may be written y^ (i/^ + 2yx^ - 2lyx + 2/3 y) By dx = 0. Jo Hence, we infer that y" + 2yx'^ - 2lyx + 2^'y = 0, so that 7f + 2x'-2lx + 2j3'==0 (1). This indicates that the generating .curve is an ellipse with the axes in the ratio of 1 to \/2. The solution however is not yet completed ; for we must shew* that the relation just found will 8TRAUCH. 393 make - = l. This we will sliew presently ; but we will previously advert to the terms which arise from a change in the limit x^ of the integrations. Suppose then that 0?^ becomes x^ + dx^ , then to the first order the following is the increment of the expression which we have to make a minimum, (I'+^'-l ^^1-^2 {fx)^dx^-\-^y^^dx^, where the subscript denotes that x is to be made equal to x^. In order that this increment may vanish, we must have either ^1 = 0, or {y- + a?-lx + ^)=0, and the latter combined with the general relation (1) leads also to y^ = 0. Thus at the lower limit the generating curve meets the axis of figure. We have now to shew that it is possible to have y^xdx /, =^' when y is determined by equation (1), and x^ is such that y vanishes when a; = 0;^. We have from (1) ^^+K^~^)=^'^'^ let a and ai^2 be the semiaxes of the ellipse determined by this equation ; then — — 2/3' = 2a^, and (1) becomes ^ f+2(x-i)^2a\ 394 SYSTEMATIC TREATISES. This equation indicates that the centre of the ellipse is at the distance— from the origin. Assume a?j = 2a — c, then I - = Xi — a = a — c» Hence we have to shew that =^^-p4 ^— = 2 ia-c) (2); when 2/'' + 2(a;-a + c)' = 2a^ We have y^ = 4ac ^ 20^^ + 4a; (a — c) — 20?'^ ; therefore I +2/V = J/ (/ + 4a;') = Uac -c' + ^xia- c)X - x' = (2ac - c')'* + 4aj (2ac - c') (a - c) + 4a;=' (a - c)' - x\ Integrate from a; = 0, to a; = 2« — c ; thus we obtain e {2a - cY + 2c{a- c) {2a - c)^ + ^ (a - c)' {2a -cf - ^ (2a - c)^ And r2a-o ^ ■, I /a;£?a;=c(2a-c)'4-^(a-c) (2a-c)^-- (2a -c)*. Thus the left-hand member of (2) becomes c' + 2c (a - c) + I (a - c)'' - ^ (2a - cf c + g (a-c)--(2a-c) that is STEAUCH. 395 and (2) becomes 5(c + 2a) 2(c^ + ac+4a'') 5{c+2a) therefore 7c^ + lac - 20^ = 0. 5{c+2a) -^ ^' This equation furnishes one positive value of -; it is ap- 3 - proximately equal to — . Then ^ is to he found in terms of a from the equation |'-2/3' = 2a^; this gives a negative value for /3', as should be the case, because from (1) we obtain 3/^ = — 2/3' when x = 0. The constant a is to be determined from the given volume, that is by means of the equation /■2a -e TT I 2[a^—(x — a + cY} dx «= the given volume. To shew that we have really obtained a minimum we should investigate the terms of the second order in the variation of - ; to this we shall now proceed. The variation of - arises partly from the change of y into y + Sy, and partly from the change of x^ into x^-\-dx^. We ^hall first shew that by reason of the suppo- sition that y vanishes when x = x^, the change in w or v arising from the change of x^ into x^-\-dx^ may be disregarded. For example, consider -y; the change in v produced by the change rivi + dxi is I y^x dx ; and as y itself is indefinitely small for values of X lying between ic^ and x^ + dx^ , the above integral may be considered of the third order of small quantities* Similar remarks 396 SYSTEMATIC TREATISES. hold with respect to the change of u. Thus to the second order 4/ we may say that the complete variation of - is ' P U Thus to the second order we obtain 7: — , where Q V and Q=l + - I i/xdi/dx-{- -\ {hyfxdx, V J Q "Jq This gives for the variation the following terms of the first order, - {f + 2yx^)Bydx--^\ yxhydx, together with the following terms of the second order, We shall denote the terms of the first order by M^ and those of the second order by il^ ; so that if the complete variation of - to the or second order be denoted by 8 - , we have V Sl = M,+M, (3). Now since the volume is to be constant we have r\y + Byydx-r'y'dx=0, Jo Jo STRAUCH. 397- that is 2 \'yBydx+ j''\Byydx = (4). •'o •'o Multiply (4) by /9 and add to (3) ; thus B- = M, + 2^f%Bydx + M, + ^r\Syydx. '^ •'o Jo And ilfj + 2/3 I yBy dx vanishes by (1) ; thus Jo Bl^M,^^\"\lyfdx, thatis ^l^\\'''(^ + ^^-l^ + ^){^yYdx - ^ i^y^ ^y dx) (^jy [Ix - /3') yBy dx) + ^^(/JV^ ^y dx)\ that is ^ I = - /^ V i^yY dx+-^ ij^'y^ ^y ^*) (f'V^y dxj . This value of S - is true to the second order, that is, no term V of the second order has been omitted. But from (4) we see that yZy dx is itself of the second order, so that the latter of the above two terms is really of the third order. Hence finally to the second order ^l-\l"y''^^)'^' and as the right-hand member of this equation is positive we have obtained a minimum value of - . V 340. The criticisms which Strauch offers on preceding writers are sometimes of a very trifling character ; we have already seen an instance in Art. 29, and we will now notice two others. 398 SYSTEMATIC TKE-ATISES. In tte problem solved by Poisson whicb we have reproduced in Art. ^9, Poisson's own result has 6 instead oiO-fA-, that is, Poissori has not explicitly introduced the constant A in his last integra- tion. Strauch refers to this slight omission in such a manner as almost to lead a reader to suppose that Poisson's investigation must be altogether unsatisfactory. See Vol. ii. page 504. On pages 747, 748 of his second volume Strauch solves a problem of a relative minimum as an example of Euler's method. Required a curve such that the area bounded by the curve the axis of X and ordinates at fixed points of this axis shall be constant, and at the same time the centre of gravity of this area at a mini- mum distance from the axis of a?. Let the abscissse of the fixed points be a and a j then -j^ is to be a minimum while I ydx is constant. •' **. J a Let 'U-~ +^L^^^ ••••(^)' 2\ y dx J a where X is a constant ; and let I ydx he denoted by A. J a I yZydx I y^ dx . Then BU=-^-^ 2^^^ I'^hy dx+L [hydx .....{2). Now put \y^dx= C ["ydx (3), J a J a then (2) may be expressed thus, BU=-^j"{2y-C + 2AL)Sydx. Thus 2y- C+2AL = (4), so that we obtain a straight line parallel to the axis of x for the required curve. Then from (3) we obtain STRAUCH. 399 [ 2 J («-«) = <^ 2 («"«)' tlierefore G = 2AL or = — 2AL ; the former by (4) gives the in- admissible result y = 0, the latter gives y = G. Now let the constant area be denoted by ff^; then since J a we obtain C{ix — a) =g^. Strauch now proceeds to investigate the terms of the second order ; he arrives at the result that the sign of these terms is the same as that of l>y'^-^a{iy^'^)' and he says that as we cannot assert that the sign of this expression is positive we are not justified in concluding by this method that there is a minimum, although it is obvious from statical consider- ations that our result does give a minimum. He therefore con- cludes that Euler's process is defective. The answer is obvious. Since the area is to be constant I By dx is absolutely zero, so that J a we are sure of a minimum from Strauch's own process. It will be found on examining Strauch's investigation of the terms of the second order that he has in efiect in one place himself recognized that I By dx is zero. The whole solution is more laborious than J a was necessary ; for since ydx is constant we might instead of J a Strauch's value of U have used the more simple value given by U= I y^dx + L I y dx, J a J a Strauch's objections to the methods of Euler and Lagrange for solving problems of relative maxima and minima seem unim- portant ; and his own method is unnecessarily complex. See Vol. I. pages 339—555, and Vol. ii. pages 740—763. 400 SYSTEMATIC TREATISES. 341. It will be convenient to notice in connexion with the work of Strauch an elaborate memoir which he presented to the Academy of Sciences at Vienna in 1856, and which may be re- garded as a continuation of his work. The title of the memoir is Anwendung des sogenannten VariationscalcuV s auf zweifache und dreifache Integrale; it was published in 1859 in the 16th volume of the Denkschriften of the Academy. The memoir occupies 156 large quarto pages, and is remarkable for the accuracy and beauty of the printing. The introduction refers to the memoirs of Delaunay, Sarrus and Cauchy, which we have described in Chapters vi, vii, Vlll. Strauch considers that these memoirs do not really effect what was required by the Academy of Sciences at Paris when they proposed their prize subject; see Art. 133. Accordingly he undertakes in the present memoir to investigate the variations of double and triple integrals. After some explanatory remarks respecting his notation he proceeds to the variation of double integrals ; this subject occupies pages 8 — 78 of the memoir. This part of the memoir contains little more than the author had already given in his work, for the most general investigation which occurs is that which we have abeady stated to be the most general investigation in his work; see Art. 336. The methods are the same as in his work; he begins with simple cases and proceeds to those which are more complex ; he gives a full account of the various suppositions which can be made respecting the limits of the integrations, although his statement of the manner in which the arbitrary functions or con- stants must be determined is too vague and general to be of much value. He usually investigates the terms of the second order, but in transforming these terms he is content with imitating the method of Legendre. The variation of triple integrals occupies pages 79 — 132 of the memoir, and is treated in his usual manner by the author. The most general investigation which is completely worked out is the variation of a triple integral in which no differential coefficient occurs of an order higher than the first; some more general investigations are partially worked out. Four problems occur as examples in this part of the memoir. The first STfiAUCH. 401 is to find w so that the following triple integral may have a maxi- mum or minimum value, III {^'-(^J&)]'^^'^^*' where ^ is a constant, and the limits of the integrations are all constants. The other three problems are modifications of that which we have given from Sarrus in Art. 180. The pages 133 — 154 of the memoir contain some remarks on the memoirs of Sarrus, Cauchy and Delaunay. Strauch quotes at full the result which Sarrus obtains for the problem which we have explained in Art. 194, and compares this result with that which he obtains by his own processes and in his own notation. Strauch gives that result from Cauchy's memoir which we have investigated in Art. 192, and compares it with that which he obtains by his own processes and in his own notation. In his remarks on Delaunay he intimates that some terms are omitted by Delaunay in his formula3 ; see pages 147 and 148 of the memoir. There is however no error in Delaunay's formulge; the terms in question do not appear because the problem which Delaunay con- siders is not the most general that could be proposed, as we have already stated in Art. 138. Again on page 149 Strauch intimates that Delaunay has only two equations for detennining certain arbitrary functions, while four are required, which he has himself supplied ; Strauch's four equations would however reduce to two in the particular case which Delaunay considers. 342. The next of the three comprehensive treatises is Mr Jellett's, entitled An elementary treatise on the Calculus of Variations hy the Rev. J. H. Jellett. Dublin 1850. It is an octavo volume of 377 pages, with a preface and introduction of 20 pages. This valuable work constitutes the only complete treatise on the Calculus of Variations in the English language, and will neces- sarily be studied by all who wish to pass beyond the rudiments of the subject. A brief outline of the work with some remarks on 26 402 SYSTEMATIC TREATISES. a few incidental points is consequently all that will "be required here. 343. The introduction contains a sketch of the history of the subject; it appears that the author had studied the memoirs of Poisson, Ostrogradsky, Jacobi and Delaunay, but had not seen that of Sarrus. The first chapter is entitled Definitions and Principles; it occupies pages 1 — 10, and explains what is meant by a variation. A very important remark occurs on page 5, " ... . many writers on the Calculus of Variations have been led into considerable difficul- ties by an unsteady use of the symbol S, a symbol which they employ sometimes to express the increment which a function receives in consequence of a change of form only, and sometimes to express the increment which it receives from the variation, not only of its form, but also of its independent variables. We shall then use the symbol S to denote that species of increment which is peculiar to the Calculus of Variations, that, namely, which a function receives in consequence of a change in its form only. We shall, as in the Differential Calculus, denote by the symbol d that incre- ment which a function receives in consequence of a change in the magnitude of its independent variables." Accordingly in Mr Jellett's work the independent variable is not supposed to undergo variation. It has already been stated in the course of the present work that this appears the best method of treating the subject. 344. The second chapter is entitled Functio7is of one indepen- dent variable; it occupies pages 11 — 30. It contains the ordinary investigations and transformations of the variation of a single in- tegi-al so far as terms of the first order, and also an investigation of the terms of the second order; the usual expression second variation is adopted for these terms, but a good note is given on page 355 respecting the ambiguity of this expression. The third chapter is entitled, Maxima and minima of indeterminate functions of one independent variable; it occupies pages 31 — 136. This chapter contains the ordinary investigation of the equation or equations which must hold in order that an integral may have a JELLETT. 403 maximum or a minimum value. Jacobi's theory for distinguisliing between a maximum and a minimum is fully developed ; the author here follows the guidance of Delaunay, see Arts. 230 — 236. This chapter contains a very important discussion as to the number of constants which can occur in the solution of a certain problem, and as to the number of them which are indeterminate. Let it be required to make the integral 1 Vdx a maximum or a minimum, J cJ7q where V contains x, y, s, and the differential coefficients of ?/ and s with respect to x ; while at the same time a relation L = is always to hold among these quantities. The following is the conclusion. Suppose that V contains ^ and its differential coefficients as far as that of the order n inclusive, and z and its differential coefficients as far as that of the order m ; suppose that the equation jL = is of the order n in differential coefficients of y and of the order m in differential coefficients of z. Then (1) If m be greater then m and n greater than n the order of the final differential equation will be the greater of the two quantities 2 (w^ + n') and 2 {m + n), and there will be a sufficient number of ancillary equations to de- termine the arbitrary constants which enter into its solution. (2) The same conclusion holds for the case in which m is greater than m and n less than n. (3) If m is greater than m and ?i greater than w, the order of the final equation will be in general 2 {m +n)-, and its solution may contain any number of indeterminate constants not exceeding the lesser of the two quantities 2 {m — m) and 2 [ri — w). Mr Jellett points out that a remark made by Poisson in the ninth section of his memoir is inconsistent with these results. 26—2 404 SYSTEMATIC TREATISES. The whole of this chapter is illustrated by examples which are fully solved. 345. The fourth chapter is entitled Application of the Calculus of Variations to Geometry . I. Theory of Curves; it occupies pages 137 — 202. This chapter consists of a collection of problems, including those of historical celebrity ; they are all fully solved. The fifth chapter is entitled On multiple Integrals in general; it occupies pages 203 — 218. The sixth chapter is entitled Functions of two or more independent variables; it occupies pages 219 — 238. The fifth and sixth chapters contain the variation of multiple inte- grals ; the methods are those of Ostrogradsky and Delaunay. The most general result obtained is equivalent to that which we have given in Art. 144 after Delaunay. The seventh chapter is en- titled On maxima and minima of functions of two or more inde- pendent variables; it occupies pages 239 — 275. This chapter illustrates and applies the results of the preceding chapter ; several examples are discussed in order to shew the treatment of the limiting equations. 346. The eighth chapter is entitled Application of the Calculus of Variations to Geometry . II. Theory of Surfaces; it occupies pages 276 — 286. The ninth chapter is entitled Application of the Calculus of Variations to Mechanics ; it occupies pages 287 — 334. This chapter besides the usual examples contains a section on the application of the Calculus of Variations to the deduction of equa- tions of equilibrium and motion. The tenth chapter is entitled Application of the Calculus of Variations to the integration of functions of one or more independent variables ; it occupies pages 335 — 354. This chapter investigates the conditions of integrdbility of various expressions. The remainder of the work consists of notes. 347. It may be of service to students into whose hands the work under consideration may come, to advert to some points which may occasion a little difficulty ; and on this ground we shall now venture to oifer some remarks. 348. In the fourth chapter of Mr Jellett's treatise many of the problems are solved by using the arc s of a curve as the inde- JELLETT. 405 pendent variable ; tlie metliod however is free from the objection stated in Art. 338. There is an example on page 138 and the following pages. In the course of the solution a constant a occurs, and it is stated that the " existence of the arbitrary constant a is an ambiguity necessarily introduced by the selection of s for the independent variable." A reason is then assigned for making a = ; but the reason does not seem satisfactory. It appears that the term yu.^ Js, — /^o ds^ is omitted in the discussion of the limiting terms on page 141. The whole expression relative to the upper limit should be then giving to m^ the same meaning as Mr Jellett does, we have 8y. + (|\*.=».{8-. + ©/4 (1). By means of (1) the expression relative to the. upper limit becomes ft*. + \ (|)_ K + \ (|)_ {». ^. + »". (|)_ ds, - (|)_ A, j . Hence ©/'"'©r" '''■ Substitute from (3) in (2) ; thus therefore ^^ = \j . This proves that a = ; since the book proves that X = /a + a. 349. We have stated in the preceding Article that it appears that /i-j ds^ — ^i^ ds^ is omitted in the discussion of the limiting terms. 406 SYSTEMATIC TREATISES. In support of this remark we may advert to Art. 152 of the present work. There by taking account of certain limiting terms we obtain the equation X' ^ dx , ■, dy , dz -3 = 1 + « -J- + & -f + c -j- p as as as this equation does not occur in Mr Jellett's investigation. The truth of this equation is confirmed in Art. 157 by its agreement with a result obtained by Delaunay. There is a difierence in the methods we have used in Arts. 152 and 348. In Art. 152 we followed the ordinary method and ascribed a variation to the independent variable s ; in Art. 348 we do not ascribe a variation to s. The final results will agree in the two methods, but the processes will difier. Thus in Art. 348, if we follow the ordinary method the whole expression relative to the upper limit will be ^^l^h + \ instead of what we have given ; and instead of (1) we shall have Thus the expression above becomes (^.-^)&.+^{(|\+™.(|)J%., and from this we obtain as before ft-X, = 0, and (|)^+». (1)^ = 0. On the other hand, suppose that in Art. 152 we follow the second method. Then instead of the term /„ dx ^ dy dz \'\ » V as as as p J P which is there given, we should have simply Vds. But now the JELLETT. 407 variations of the limiting co-ordinates will not be simply Bx, hy, tz, as in Art. 152, but hx + -r-ds, Bi/ + -4-ds, Bz+-j-ds respectively ; and these must vanish at the limits, since the limits are supposed fixed. Thus we shall obtain finally the same result as before. Of the two methods which can be used, Mr Jellett has decided in favour of that which does not ascribe a variation to the indepen- dent variable, see Art. 343. But It would seem that In the fourth chapter of his work he has not adopted uniformly the consec[uence3 which follow from this decision. 350. Remarks similar to those abeady made apply with respect to pages 153, 155, 178, 181, 183 and 299 of the book. Again, on page 170 it is remarked, " and the remaining con- stant, a, depending upon the given length of the curve...." Nothing however has been previously said respecting the given length ; and it appears here as before that /x^ ds^ — jx^ ds^ should be added to the limiting terms if we adopt the method of Art. 349. Or if we adopt the method of Art. 152 we must add (/^ - X + /A»^ S5j - (^ - X + /*», &o. Again, on page 175 it is stated, " the superfluous constant a will be determined by expressing the area as a function of that constant and equating its differential to zero." This reference to the ordinary Differential Calculus is unnecessary ; for the Calculus of Variations supplies sufficient conditions for determining the con- stants. The problem under discussion Is, to find a curve of given length such that the area bounded by the curve itself, its two extreme radii of curvature, and the arc of the evolute between them may be a minimum. This problem is solved in most elementary treatises, and the result obtained Is that the curve must be a cycloid; this result Is obtained by the ordinary processes of the Calculus of Variations. In fact if we adopt the method of Art. 152 408 SYSTEMATIC TREATISES. we shall find that the following limiting terms have been omitted in the hook, where /j.' = 1 and fx = p+ a, constant. From considering these we find that we must have Xj = 0, and \ = 0, since p^ and p^ vanish. Then by page 168 af the work, we have dx , /dx\ /dx\ for it is shewn on page 169 that & = 0. Thus f-j-j and l-r-) must /dx\ vanish. Then bj page 174 since 3/1 = and [-rj = we have e=0; and this is the result which is established in the book bj appealing to the Difierential Calculus. 351. On page 165 some results are given without demon- stration. The results refer to a segment of a sphere which is required to have a maximum or minimum volume, while the surface is given. Let a denote the radius of the sphere, h the height of the segment, then the volume of the segment is tt iah^ — ^j . Since the surface is given, ah is equal to a constant, which we will denote by Jc^. Let 1/ denote the radius of the plane base of the segment ; then therefore h^ = 2k^ - y^. Thus the volume = tt -fo V(2A;'' - /) - ^^^'~^'^ | = Fsuppose. Now y is supposed to be an ordinate of a given curve, and V is to be made a maximum or a minimum by properly choosing this ordinate. Let x denote the abscissa corresponding to the ordinate y. Then we have JELLETT. 409 therefore ^^- ^3/(^^-3/') % tneretore ^^ - ^(2A;2_^2) ^^. "We have now three cases to examine, namely (1) \i y itself be a maximum or minimum F will he a maxi- mum or minimum respectivelj provided Ti — 3/^ he positive, and a minimum or maximum respectively provided H — 'if' he negative. (2) The value y^h makes -j- zero, and makes -j-g- negative civ provided -j- he not zero ; thus in this case F is a maximum. If y is dV itself a maximum or minimum when y = h, then 3— changes sign when y = h, and so F is itself a maximum or minimum respect- ively. (3) With respect to the case of 3/ = we must remark that the question does not suppose that y is capable of becoming negative. If the given curve touches the axis of x then the value y = occurs simultaneously with ;^ = 0, so that y is then a mini- mum and so is F. These results do not agree with those in the book. The case in which y = h seems there overlooked. li y = h we have h = Jc=:a. And it may be seen that the relation on the 14th line of page 165 of the book may be satisfied by supposing a = y and the angle CFY zero. 352. On page 365 the following problem is suggested ; to con- struct upon a given base a curve such that the superficial area of the sm-face generated by its revolution round AB may be given, and that its solid content may be a maximum. 410 SYSTEMATIC TREATISES. Take tlie axis of x as tliat of revolution ; then adopting the usual notation we require that tt Wdx should be a maximum while 27r \y V(l +p^) dx is given, the limits of a? being supposed fixed. Thus if a be a constant we have to find the maximum value of /I= Hence we must have y + aV(l+/) = ^ /^ ^ ■2^ (1); this we know leads to 3/^ + 2a3,V(l+/) = -.f^ + 5, V(l+/) therefore -—^^^—- = l-f (2), where & is a constant. Then since y is to vanish at the two fixed points we have 5 = 0, and then by completing the solution we obtain a semicircle for the required curve, and therefore a sphere for the solid generated. Mr Jellett points out that this solution is unsatisfactory, because the superficial area of a sphere described upon a given diameter is a determinate function of that diameter, and cannot therefore be made equal to any given quantity. Mr Jellett proceeds to remark that the process of the Calculus of Variations fails in this case. We suggest the following as a solution of the problem. Let the figure A CEDB consist of two straight lines A 0, BD perpendicular to the axis of x, and of the arc GED which satisfies the differential equation (2) ; see figure 10. Take A as the origin ; l^i AC = y,,BD = y„AB = x^. Then the volume of the figure formed by the revolution of A CEDB round AB is tt / y^dx; and the surface, including the circular ends, is •^2/1' + "^l/o + 27r r^y V(l +/) dX' JELLETT. 411 Now suppose that y is changed into y + By, then the variation of the volume is 27r I y Sy dx, and we have to make this zero for •'0 such variations as leave the surface unchanged ; that is, for such variations as make 27ry^By^ + 27ry,By, + 2'irB\%>^{l+p')dx=0 (3). Jo Thus if a represent a constant we must make The part under the integral sign vanishes hecause we suppose equation (1) satisfied. So that we only require in addition I .,/ ,, - ll = 0, and 1-777^— ir + ll = 0. V(i+/) Jo [V(i+/) h This leads to p^ = + cc and p^ = — ao- that is, the curve must join on continuously to the straight lines at C and D. Then it appears from (2) that y^ = i when p is infinite, so that AC = BD. The constants a and h, and that which would arise from in- tegrating (2), must then be determined so that y^ = h when x=0 and when x = x^, and that the surface may have the given value. Suppose however the circular ends are not to he included in the given surface. In this case y = — a furnishes a solution. For the terms ay^ By^ + ay^ By^ do not now occur in (4) ; and the value y — ~a makes vanish, and it gives » = so that ■ ,1 o, also vanishes. Thus / V(l +/) we obtain a cylindrical surface ; a will of course be negative, and will be determined by the condition that — 27r ax^ must be equal to the given surface. 412 SYSTEMATIC TREATISES. 353. A mistake occurs on page 376 of the book whicli may be noticed. The integral I tan^ sin 6 dd is made equal to a finite ''0 negative value, the fact being overlooked that tan^^ becomes infinite between the limits of integration. And the same mistake occurs on page 377 where the integral I 2 f/-, 2\ is taken between limits which include p = and make the integral really infinite. In concluding we may strongly recommend the student of the Calculus of Variations to master this important volume. A trans- lation of it into German has been advertised, but the present writer has not had the opportunity of consulting it. 354. The last of the three comprehensive treatises is by Dr Stegmann, entitled Lehrhuch der Variationsrechnung und Hirer Anwendung hei Untersuchungen iiber das Maximum und Minimum. Kassel, 1854. It is an octavo volume of 417 pages with a preface of 16 pages. In the preface the author states that he had long been of opinion that the Calculus of Variations was treated in a meagre and un- satisfactory manner in elementary treatises, and had resolved to undertake the task of producing a more complete work on the subject. The work of Strauch had not appeared when first this resolution was formed; after it was published the question arose with Stegmann whether he should continue his design, since he had no intention of ofi^ring to his readers such a rich collection of problems as Strauch had supplied. Ultimately he resolved to complete his original design. In addition to the works of Dirksen, Ohm and Strauch, Steg- mann refers to the memoirs of Poisson and Ostrogradsky. He has discussed numerous problems as illustrations of his theory, but he does not present his work as a collection of problems, for the development of the general theory has been his main object. In solving his problems he has imitated Ohm and Strauch in investi- gating the terms of the second order so as to discriminate between maxima and minima values. STEGMANN. 413 355. The work consists of six chapters and two supplements. ^)^^« J a. J a (8) The curve down which a body must fall in a resisting medium so as to acquire the greatest velocity. (9) To find the minimum value of 1 (-^) dx, under the conditions thatyo=l and that ydx — — y^, (10) The problem we have enunciated in para- /, graph (3) of Art. 311 ; Stegmann does not however allude to the difficulty which occurs in the particular case which we have examined in Art. 352. 359. The fifth chapter is entitled On Mixed Variations with .simidtaneous changes of the independent variable; it occupies pages 265—327. In all the investigations hitherto given in the book the limits of the integrations have been supposed fixed and the independent variable unsusceptible of variation ; Stegmann proceeds in the pre- sent chapter to give that extension to his formulas which they require in order to apply to problems in which the initial and final values of all the quantities which occur are changed. He now adopts the common method of ascribing a variation to the indepen- dent variable. Suppose x the independent variable and y the dependent variable, let these become by variation x + hx and y + By respectively ; then Stegmann obtains a relation denoted thus By = (8) y +pBx. This result might be presented as a definition, namely, let By — pBx be denoted by {B)y, and then it might of course be considered absolutely true. Stegmann however adopts a different 27 418 SYSTEMATIC TEEATISES. method ; lie defines (8) y and by means of geometrical considerations establishes the truth of the relation as far as the first order of small quantities. It is then necessary for him to shew that where 9. — 'j^~ 'i^ 5 ^^^ generally that d^y_d^{h)y d^y ^ dx""' dx"" ^ dx""^'^"^' His method is the following, s _^dy _dx dhy — dy dSx _ dSy dBx ^ ^~ dx~~ dx^ dx " dx ^ put (8)2/ ■\-'pZx for 82/ and qdx for djp^ thus this may be written Z]^ = {^)'p ■\- qBx. Stegmann subsequently gives the common geometrical illus- tration of the relation Sdx = dSx. The above investigation of the value of Bp cannot be regarded as absolutely true, but only as true to the first order. Suppose now that U= I Vdx, and that the variation of U is J a required ; Stegmann proves that the result obtained when x was supposed unsusceptible of variation, so far as terms of the first order are involved, requires only the following modifications ; By, Bp, ... have to be changed into (8)y, {B) p, .... respectively, and the following limiting terms added, V^B^— VaBa. Two proofs are given of this statement. The formulse are illustrated by discussing the problem of the brachistochrone in the case where there is no resistance, and also in the case where there is, and the problem of the shortest line. In STEGMANN. 419 both these problems various suppositions are made with respect to the limiting conditions and carefully examined. For example, take the problem we have considered in Art. 300 ; Stegmann adopts the suppositions there made and arrives at the results there ob- tained by interpreting the terms of the first order. Then he makes another supposition ; let the limiting values a?^ and x^ be connected by the relation «2 — a;^ = a constant, then dx^ = dx^ , and instead of the two equations obtained by equating to zero the coefficients of dx^ and dx^ we have now the single equation ^{l+f)^{h + x-xj], W{l+f)V{h+x-x,)l'^l,,dx/'' ''' this reduces to {p^lr' {x)+ll (px{x) + ll 1/1 1 therefore i|r' (a;J = x {xj ; thus the tangents to the limiting curves at the points where the described curve meets them are parallel. Stegmann also considers briefly the subject of the discrimina- tion between maxima and minima values when the independent variable is supposed to undergo a variation. Here of course allow- ance has to be made for the circumstance that some of the formulas employed were only true to the first order. He illustrates his remarks by considering the problem of the shortest line between a given point and a given curve. On the whole the chapter appears to be a good exhibition of the method which the author selects, but the method seems far less simple and satisfactory than that of not allowing the inde- pendent variable to undergo variation, but obtaining the ' requisite generality by changing the limits of the integrations. Two other subjects may be mentioned which are Introduced into this chapter. On pages 278, 279 Stegmann proves the theorem 27—2 420 SYSTEMATIC TREATISES. wliicli we liave expressed in Art. 93 thus, Hy + Kz = ; the proof does not depend on the Calculus of Variations. On page 292 Stegmann considers the case in which the function under the in- tegral sign may itself involve the limiting values of the variables or differential coefficients ; he points out however that the limiting values of the highest differential coefficient when there is only one dependent variable must not occur ; because if in such a case we wish to make the integral a maximum or a minimum we have in general more conditions than disposable quantities. A similar remark holds when there is more than one dependent variable. 360. A supplement to the third, fourth, and fifth chapters occupies pages 327 — 338 ; it draws attention to the method of solving problems in this subject which was adopted by the early writers, and refers to the memoir of Schellbach. Stegmann solves two problems by this method. (1) To find among all curves of given length that for which F{^y) dx is a maximum or a mini- J a mum. (2) The shortest line on a surface of revolution. 361. The sixth chapter is entitled, On the variations of func- tions of two independent variables; it occupies pages 338 — 395. On page 11 of his work Stegmann seems to indicate that mixed variations occur only in the fifth chapter, but we find them again in the first section of the sixth chapter. Suppose z any function of x and y, say z =f{x, y) ; let [x, y, t) denote any function of x, y, and t, which reduces to f{x, y) when t vanishes. In j> {x, y, t) change x into x-\-hx, y into y + hy, and t into t + ht; then a result is obtained which is denoted thus, or by supposing i = 0, S^ = (S)^ + ||S«' + |Sy- STEGMANN. 421 Then since this is true whatever function of x and y is denoted by z^ Stegmann says we have ^ dx dof dxdy ^ _ d{B) z d^z ^ d^z ^ dy dx dy dy^ "' and so on. This method seems however an unsatisfactory proof of these formulae; see Arts. 102 and 124. Stegmann next refers to questions similar to that in Art. 3, hut involving more than one independent variable. He solves the following problem ; to determine a surface having the property that the sum of the squares of the intercepts cut off from the co-ordinate axes by the tangent plane at any point shall be a minimum. Thus in the usual notation {fdxy where ^ involves x, y, and the dififerential coefficients of y ; before reducing jBcjidx in the ordinary way by integration by parts, Stegmann makes some remarks on the attempt to solve the problem 424 SYSTEMATIC TREATISES. •whicli is made by supposing S^ = ; see his page 85. The relation B(f> = can only indicate one of these two things, either (f> does not change by ascribing a variation to y and its differential coefficients, or else ^ is itself a maximum or minimum. The former supposition is impossible, since With respect to the latter supposition it is to be observed that if be itself a maximum or minimum for all values of x between given limits, then I <^ dx will also be a maximum or minimum re- spectively, the integral being taken between those limits. This Stegmann proves by means of a figure which is constructed by taking the ordinate of a curve always equal to ^. The proof amounts to the consideration that the integral must be a maxi- mum or a minimum, because each of the elements of which it may be ultimately regarded as the sum is a maximum or minimum re- spectively. It is however not true conversely that any relation which renders \ ^dx a maximum or minimum will make ^ also J a such for all values of x between a and ^. This is illustrated by a figure which amounts to the consideration that I (p^ dx may be greater than I ^ are J a less than the corresponding values of (p^] for other values of ^^ may be greater than the corresponding values of 02 . Thus the conclusion is that the relation B(f> = will not neces- sarily supply all possible solutions of the problem of finding the maximum or minimum value of \(f)dx. 364. On page 109 of his work Stegmann makes a remark which relates to the use of a series to represent a variation instead of a single term ; see Art. 334. Stegmann is investigating the maximum or minimum of I / ^^' ^^^ ordinary mode would be to change ^ into 'p + Sp, and then to examine the terms STEGMANN. 425 involving Sp and {BpY, But suppose we change p not into p+Bp but into a series, after the manner of Straueh ; let this series be Arrange the variation of the proposed integral according to powers of k ; thus we obtain for the variation i. ■/x^/(l +/) + 2 J. L 1.+/ +^^ ^"^^J dx + ... Va;V(i+/) In order that there may be a maximum or minimum we must have in the usual way ~ = a constant. VajV(l+/) Stegmann then remarks that we are prevented from ascertaining what the sign of the term involving /c^ is, by reason of the presence of '^'(a;) which is altogether independent of II' (a?) . He does not notice that the relation which has been already assumed in order to make the coefficient of k vanish, also makes the coefficient of "^'{x) constant in the term involving k^; hence it will be found that since the limiting terms of the first order are made to vanish, the terms of the second order which depend on "¥ 'x will also vanish. It is in fact this circumstance that renders it useless to adopt the form of a series instead of a single term in order to denote a variation. 365. On page 140 of his work Stegmann is discussing the problem of finding the curve which with its evolute includes a minimum area. Let ?7=f*il±^'rf.; Ja q by making the terms of the first order vanish in the variation of U we obtain a cycloid for the curve. The terms of the second order may be put in the form £ {2 (1 +/) {BpY +(2pBp- i±i^' Bq dx 9. 426 SYSTEMATIC TREATISES. and Stegmann says that neither - nor p can become infinite be- tween the limits of integration ; so that the solution he has obtained gives a minimum. But jp is infinite at the cusps of the cycloid, and thus Stegmann is wrong. But although p is infinite yet (1 -I- «2\f -^ =^--^ does not become infinite, this being the radius of curvature of the curve : hence h fortiori ~ , ^— , — 5-^^-^ and 1-^ do not become infinite. Thus if h^ and hg^ are indefinitely small throughout the limits of the integration the quantity under the integral sign in the above expression will not become infinite ; so that the result obtained is really a minimum in comparison with all adjacent curves which can be obtained under the limitation that hp and hq shall be indefinitely small. With respect to the problem in question it will be useful to notice the conclusions of other writers. Thus in De Morgan's Differential Calculus, page 463, the following stateinent is made, " the radii of curvature at the extreme points are both =0; which in the cycloid only happens at the cusps. Hence if ^ and B be the given points, every such figure as that in the diagram gives an algebraical minimum : that is to say, any slight variation of the upper curves with a corresponding variation of the lower evolutes would increase the area contained. There is no absolute arithmetical minimum; for by sufficiently increasing the number of revolutions of the generating circle we might diminish the whole area without limit." The diagram referred to supposes the generating circle to have turned round three times completely, so that there are three complete arcs of a cycloid between the two fixed points. There is no investigation of the terms of the second order to shew that any slight variation would increase the area. The problem is solved by Strauch, and he exhibits the terms of the second order, but makes no remarks of importance. See his Vol. II. pages 289—291. Mr Jellett discusses the problem, and makes some remarks on the result; see pages 172 and 177 of his work. He gives a STEGMANN. 427 figure consisting of a single complete arc of a cycloid with its extremities at the two fixed points ; the two fixed points are also connected bj a curve which is composed of two complete arcs of a cycloid, one of which may if we please be supposed indefinitely small, and the other finite and difiering infinitesimally from the single complete arc first considered. It is easy to shew that the area in the second case is less than the area in the first case ; nevertheless the first is to be considered a real minimum in the proper sense of that term, because the second curve cannot be deduced from the first by a legitimate variation. 366. In Art. 202 we have referred to a result obtained by Legendre in discussing the following problem ; required to connect two fixed points by a curve of given length so that the area bounded by the curve, the ordinates of the fixed points, and the axis of abscissae shall be a maximum. Stegmann discusses this problem and an-ives at the same results as Legendre, though he does not refer to him ; see Stegmann's work, pages 175 — 180. Let \ , h^ be the co-ordinates of one of the fixed points, which we will denote by A ; let \, \ be the co-ordinates of the other fixed point, which we will denote by B ; and we will suppose \ less than \, and h^ less than \. Then with the usual notation I ydx is to be a maximum while I /v/(l +i^^) <^^ is to have a given value. J hi Then we proceed to make (3/ + \V(l+i5^)M^a maximum where J hi X is a constant. Therefore l_x^-^ = (1), therefore x— C,= ,,, ^ .,, , ' v(i+y) dy _ X— 0^ therefoie therefore y- C, = + ^l[\^-{x - C,f] (2). 428 SYSTEMATIC TKEATISES. It is easy to see that the sign of the terms of the second order is the same as that of rhg {BpY dx. and is therefore the same as the sign of X. Then (2) gives for the required curve an arc of a circle of which X,^ is the square of the radius, and from (1) it may he shewn that this arc will be concave to the axis of aj if \ he negative ; so that an arc of a circle concave to the axis of x gives a maximum area. The constants X, G^ , and Cj are to be determined by making the arc go through the points A and JB and have the given length. This given length must of course be greater than the straight line which joins A and JB. The solution thus obtained is satisfactory as long as the concave circular arc joining A and B falls entirely between the lines drawn through A and B perpendicular to the axis of x ; the extreme ad- missible case is that in which the ordinate at A is the tangent to the circular arc at A. Supposing then that the given length exceeds that which cor- responds to the extreme admissible case just referred to, we must modify the problem. Let the ordinate at A be produced through A to a point distant 7/^ from the axis of x ; and let the straight line of length yi — \ be considered part of the curve connecting A fhz and B. Thus we now propose to make I i/dx a, maximum while J hi ^j— ^^+ I ^/(l +j?^) dx has a given value. No change is thus re- J hi quired in the solution of the problem except so far as relates to the terms at the limits ; these formerly vanished because the extreme points were both fixed. Now we have corresponding to the lower limit the expression and to make this vanish we must have therefore p^= co . STEaMANN. 429 This requires tlie circular arc to have its tangent at the point {\, y^ where it joins the ordinate produced through A, coincident with that ordinate produced. Thus y^^ G^, and G^ — \=\\iQ radius of the circle ; and the constants will he found from these relations combined with the conditions that the circle shall pass through B, and that the length of the circular arc together with y^ — \ shall he equal to the given length. In this case then the required curve is made up of a straight line of the length y^ — \ and of an arc of a circle. The solution thus obtained is satisfactory so long as the concave circular arc is not cut by the ordinate at B produced through B', the extreme admissible case is that in which the ordinate at B is the tangent to the circular arc at B. Supposing then that the given length exceeds that which cor- responds to the extreme admissible case just referred to, we must again modify the problem. Let the ordinate at B be produced to a point distant y^ from the axis of a?, and let the straight line of length y^ — \ as well as the straight line of length y^ — \ be considered part of the curve connecting A and B, Thus we now propose to make ydx a maximum, while / yi-h+y^-K+L V(i +/) dx J hi has a given value. No change is thus required in the solution of the problem except so far as relates to the terms at the limits. We now have the expression and to make this vanish we must have therefore Pi = ^ t ^^nd p^ = — co. 430 SYSTEMATIC TEEATISES. This requires the circular arc to have its tangent at the point (A^, y^ where it joins the ordinate produced through A, coincident with that ordinate produced; and also its tangent at the point (^2' y^ where it joins the ordinate produced through B, coincident with that ordinate produced. This requires the circular arc to be a semicircle, so that y^ = y^= C^', and 0^= -^{\-\-Ti^, and the radius of the circle = - [Ji^ — h^. The constant C^ is to be found from the condition that the sum of the length of the circular arc and y^—\ and y^ — \ is to be equal to the given length. In this case then the required curve consists of a semicircular arc and two straight lines. 367. In his fourth Chapter, pages 222 — 227, Stegmann gives an investigation of the number of the constants which can occur in the solution of a certain problem, and of the number of the equations which serve to determine these constants ; see Art. 344. Stegmann' s conclusion is that in general these constants can all be determined; he does not shew that the auxiliary equations may diminish in number in certain cases, and thus some of the con- stants remain indeterminate. He draws attention however to some exceptional cases, in which the number of the constants may be less than the general theory indicates. Take for example the first case considered in Art. 273 ; here no arbitrary constants occur in the solution, so that the terms which relate to the limits must be supposed to vanish of themselves, or they will not vanish at all. In other words, if we use geometrical language, the limiting points must be supposed fixed through which the curve is to be drawn. 368. On his pages 245 — 247, Stegmann solves a problem which we will here notice. Let TJ— \ Z'"dx, where Z=\\l{\+p'')dx- J a STEGMANN. 431 required to find tlie value of y whicli makes TJ a maximum or minimimi. Here we have as far as terms of the second order . ia [V(l +/) 2(l+2?'f J and SCr=[^ jnZ"-^ SZ+ !L(!!Ji11 ^-^ {^Z)' ^ ..\dx. The investigation of Art. 38 may be applied to this problem. The quantity there denoted by v is here denoted by Z, and L = nZ^-\ AlsoP^= -^ , while iV, N\ P, Q, Q',... are zero. Thus we obtain (^ — /) P' = a constant ; and as -4 — / vanishes when x has its superior limiting value, the constant must vanish ; this leads to P'= 0, so that^ = 0. The inte- grated part of the variation also vanishes since the above constant vanishes. Since ^ = the only term of the second order which remains in S C/" is l^ Z'^-'if' {hpYdJ^-dx, irz' which is positive, and so we obtain a minimum. Stegmann's solution is effected by the use of an arbitrary multi- plier, and leads to the same result. In discriminating however between a maximum and a minimum, he retains the term M!?_zl) l^ Z''-' {hZf dx, 2 J a and this leads him to make the supposition that n is positive and not less than unity, in order to ensure a minimum. But as p is zero, hZ is itself of the second order of small quantities, and thus the term just expressed is of the fourth order, and therefore does not 'require to be retained. It will be observed that the solution ^ = can only apply when the limiting values of y are either not given or are given equal. 432 SYSTEMATIC TREATISES. When the limiting values of y are given and unequal suppose these values to be yS and 17 corresponding to the values a and ^ of x. Then putting the problem into a geometrical form, the curve re- quired appears to be made up of the straight line "which joins the point (a, yS) with the point (f, /9), and the straight line which joins the point (f , /3) with the point (f, 97) ; or at least the nearer we ap- proach to this limit the smaller does TJ become. This problem is taken from Euler's Meihodus Inveniendi ... page 94. Euler considers any function of Z instead of Z"^, and he arrives at the result j? = as necessary for a maximum or a minimum. 369. An example of a relative minimum is solved by Steg- mann on his pages 255 — 258, which we will give here. !Re- 1 r^ quired the minimum value of - I p^ dx under the following con- ditions ; yo=l (1), \'y-dx=^- 1 (2). Let \ be a constant, and let then to the first order Hence ^ - ^ = (3) y^ ax ^ and Pi~^j y^^ = ^ (4). From (3) we have therefore — [x - A)^ = 'p ■\- B^ where A and B are constants. STEGMANN. 433 The condition (1) gives therefore 2/= 1 + x—{p(?—2Ax) (5). The condition (2) gives 1+6^-1^=-^' <"'• From (2) and (4) we ohtain ^+^=0, that is -(l-^)+-=0, yx Vx Vx that is -(2-^)=0 (7). Vx By putting oj = 1 in (5), we obtain yx=l+^(l-2^) (8). The solution \ = of (7) is inadmissible, for that would make y = \ by (5), and then (2) would not be satisfied. Hence we deduce A = 1 fi-om (7), and then from (6) and (8) we deduce 12 2 ^"=-49' ^^ = -7- Now to determine whether a minimum is thus obtained, we must form the expression for hJJ correct to the second order; now we have exactly Jo 12 ^-^ ^' 2^ 2/i + Syi y^) and therefore to the second order. = {g (8i>r-^ + ^}<^-. b7 (3) ana (4). 28 434 SYSTEMATIC TREATISES. Bj integrating by parts we Have thus ^^=f {I (¥)' + ^ %1 ^^^ A, 1 f^ Now — ^ = — 3, and — \ i/dx = —l; thus finally so that we have ohtained a minimum. 370. Stegmann gives on his page 395 one application of the formulge relating to double integrals which we will reproduce. Suppose we have to find a surface of minimum area under the condition that the length of the boundary is given. The equations which must hold at the boundary are the first two of equations (18) of Art. 114. Here F=V(l + ^'^ + ^;), X=l, F=^, 7 _ d fdx\ _ d^x ds 7, _ ^ (^y\ — ^^y ^^ dz \ds) ds^ dz ' dz \ds) ~ ds^ dz ' Thus the equations are z,dx-z'dy (l^ ' + .,Vc|@ + .,g) = (1), , zdx — z'dy d^z , _ . /o\ ''"'I vci+^vo'^g ^^' and this is tlie form in which Stegmann gives them. He now takes another condition, namely that z shall he constant round the boundary, so that round the boundary dz — z dx-\- z^dy = (3), and since z is constant round the boundary (2) gives z^dx — zdy — — ... (4). From (3) and (4) we have round the boundary g;' = and z^ = 0. Also (1) becomes round the boundary ^ "" ds^ dx^ • ^^>' From (5) by integration we obtain the equation to a circle of which the radius is numerically equal to c ; that is, the projection of the boundary on the plane of [x, y) is a circle. Then Stegmann observes that this cannot give a minimum area but a maximum area, since the boundary is supposed to be a closed curve. But the result may be made useful by modifying the problem. The modification appears to be that the projection of the boundary shall be a four-sided figure having for two of its sides fixed sti'aight lines perpendicular to the axis of x, and the other two sides remaining to be determined and each being of given length. Then Stegmann says these other two sides should be arcs of circles with their convexities turned towards each other. 28—2 CHAPTER XIV. MINOR TKEATISES. 371. This chapter is intended to give an account of the minor treatises on the Calculus of Variations. It includes all the separate works which have come to the writer's knowledge, hut does not attempt to notice every case in which a chapter has been devoted to this subject in the course of a general work on analysis. A few such cases have been however included in the present list. 372. Brunacci. A treatise on the Calculus of Variations occupies pages 166 — 255 of the fourth volume of Brunacci's Corso di Matematica Sublime. Florence, 1808. Brunacci begins with some general remarks similar to those which we have given in Art. 363 after Stegmann. He considers the case in which F{x^ y) is to be a maximum or minimum by the variation of 3/, and then the case in which F{x, y, p) is to be a maximum or minimum by the variation of y and p or of one of them ; and he gives Lagrange's example ; see Art. 3. He makes some brief remarks on the history of the subject, and states that Lagrange had finally relieved it from any consideration of infini- tesimal quantities; he proposes to follow Lagrange's method in discussing the subject. He does not use the symbol ^y, but io> instead, where i is supposed indefinitely small and co an arbitrary function. In finding the maximum or minimum value of an integral \^dx he first supposes that ^ contains only x and y ; he illustrates BEUNACCI. 437 this b J two examples taken from Euler's ilie^/icx^ws Inveniendi . . . , pages 39 and 40, and in the second example he agrees with Dirksen in distinguishing between a maximum and a minimum more care- fiillj than Euler did ; see Art. 52, and Dirksen, page 202. He next supposes that ^ is a function of a:;, y, and^, and that l^c^aj is to be made a maximum or minimum; this case he illustrates hj dis- cussing the problems of the shortest line and the brachistochrone. He insists on the propriety of separating the problems which occur into two parts, one depending strictly on the Calculus of Variations and the other on the Differential Calculus; see Arts. 90 and 91. He says that this idea was communicated to him by a distinguished scholar and mathematician Paradisi, and that Euler himself would have judged it worthy of his own immortal work, the Methodus Inveniendi Accordingly Brunacci in treating the problem of the brachistochrone between two given cm-ves first supposes the extreme points fixed and obtains a cycloid by the Calculus of Variations as the required curve; then he determines by the Differential Calculus the position which the cycloid must have when its ends are supposed moveable on two cm-ves; in spite of Brunacci' s opinion his process seems longer and not clearer than that usually given which depends on the Calculus of Variations solely. Brunacci next supposes that ^ is a function of a?, y, p, and q, and that the maximum or minimum of 1 dx is required ; this he illustrates by examples drawn from pages 61 and 247 of the Methodus Inveniendi Brunacci supplies investigations of the terms of the second order for distinguishing between maxima and minima values ; he repeats the investigation to which we have alluded in Art. 216 ; he says however that it is now presented in a better form. Brunacci gives some account of problems of relative maxima and minima, and considers a few simple examples. With respect to the variation of double integrals he gives an investigation which is correct so far as it goes ; see Art. 29. He applies the result to obtain the differential equation to the surface 438 BEUNACCI. wliich is a minimum among those wliicli include the same volume. He sajs however that owing to the difficulty of integrating partial differential equations, to the difficulty of determining the arbitrary functions which occur in the solutions, and to other difficulties which arise from the nature of the problems, very little can be effected in this part of the subject; in his own words "... siamo sopra una spiaggia da cui si scopre un mar senza tine, e non ci h dato per anche d'inoltrarvisi, onde fare delle scoperte." It would appear from his page 248 that Brunacci considered that his treatise on the Calculus of Variations might be conti*asted favourably with those which had been previously published. It is not however very accurate in language or investigation ; we have already in Art. 208 pointed out an objectionable statement, and we will now indicate some others. Brunacci says on his page 168 that I f{x) dx is the sum of all possible values oi f{x) between those J a which correspond to a:; = a and x=h', this amounts to overlook- rh ing the dx which occurs in the symbol / f{x) dx. On his J a page 245 he interprets the equation xy = Sz^ to mean that the vertical ordinate is a third of the rectangle of the horizontal co-ordi- nates, instead of saying that the square of the vertical ordinate is so ; here he had previously given the statement correctly. On his page 229 he discusses the maximum or minimum of l-ylrdx, where i/r is a function of Z, and Z=^w{l-\-p^) dx. We have already considered a case of this problem in Art. 368. Brunacci by an obscure method arrives at a differential equation, and he shews that when a certain constant c' vanishes the solution is ^ = ; but this he says is only a particular solution. It will be seen, however, on his page 230 that he requires a to vanish at the limits, and ^_ cV(l+/) P so that his solution leads necessarily to c' = 0. LACEOIX, GERGONNE. 439 373. Lacroix. An elementary treatise on the Differential and Integral Calculus, hy S. F. Lacroix. Translated from the Frencli. Cambridge, 1816. This work contains a brief treatise on the Calculus of Varia- tions on pages 436 — 463, and 706 — 711. The treatise has been described with great justice as " singularly confused and unin- telligible." 374. Gergonne. Gexgorme's Annales de MatJiematiques ...,,. Vol. 13, 1822, pages 1—93. This memoir is on the investigation of the maxima and minima of undetermined integral formuloe. Gergonne considers that with many persons the Calculus of Variations is merely a mechanical process of which they do not comprehend the spirit. He proposes to shew that the questions of maxima and minima for which this Calculus was principally invented can be treated in the clearest and briefest manner by the principles of the ordinary Differential Cal- culus. He does not use the distinctive notation of the Calculus of Variations; thus for what is usually denoted by Sy he puts lY, where t is an indefinitely small quantity and Y is an arbitrary function. This memoir seems of no great use ; any student who could understand it could understand the ordinary exhibitions of the Calculus of Variations. The distinctive notation of the Calculus of Variations has always been considered one of its great advantages, and nothing is gained by discarding this notation. There are also passages in this memoir which would probably appear more difficult to a beginner than the corresponding passages in the ordinary treatises. Thus, for example, we may refer to the way in which Gergonne shews that the integrated and the unintegrated part of the variation of an integral must separately vanish in order that the integral may be a maximum or a minimum. The memoir is written with remarkable diffuseness. As an instance the following may be noticed. When Gergonne is dis- 440 GEEGONNE. AMPEEE. cussing the question of the shortest line he obtains these two equations, A "i =0 - ^ =0 and instead of inferring at once that -77-— — 75 ?2\ = a constant, and —jj- — ^ tot = a constant, he devotes a page to performing the differentiations first and then retracing his steps bj integTation ; and he makes a temporary mis- take in the course of his process by omitting \ in the fifth line of his page 3&. Page 89 is quite wrong ; the equations in the fourth line are false, since they ought to involve the partial differ- ential coefficients of ^S'; the equations given by Gergonne would make the osculating plane of the curve perpendicular to its tan- gent. The following paragraph forms the last of Gergonne's memoir. In conclusion we must ask the indulgence of the reader for the numerous imperfections and even errors which may be found in this memoir. If we may believe what is stated by Dr Prompt in a small treatise published in 1820, the work even of the illustrious Lagrange on this subject is not free from objections. The em- barrassing notation of that great mathematician on the one hand, and the brevity of Dr Prompt on the other hand, have prevented us from ascertaining to what extent these objections are well founded ; but this is a point to which we will return on another occasion. [It does not appear that Gergonne ever returned to the subject. The present writer has not seen any other notice of Dr Prompt's work.] 375. Ampere. Gergonne's Annates de Mathematiques Vol. 16, 1825, pages 133—167. This memoir is an exposition of the principles of the Calculus of Variations, and is said to have been drawn up by Ampere for his AMPERE. VERDAM, VERHULST. 441 course of lectures at the Polyteclinique School. It constitutes such an elementary treatise on the Calculus of Variations as is frequently given in works on the Differential and Integral Calculus, and presents no peculiarity. After establishing the formula for the variation of an integral Ampere shews that in order that the integral may be a maximum or a minimum the two parts of the variation must separately vanish. This he shews by supposing in the first place that the limiting values of the variables and of the differential coefficients are given ; then the part of the variation which remains under the integral sign must vanish be- cause the other part vanishes of itself. Next he supposes that the limiting values are not given ; still it is in our power to sup- pose such a variation as leaves the limiting values unchanged, and this variation must be zero, so that, as before, the part under the integral sign must vanish. Gergonne himself says in a note that Ampere is the first who has shewn distinctly that the part of the variation which is under the integral sign must separately vanish, and he admits that his own memoir was unsatisfactory on this point. 376. Verdam and Verhulst. The subject of maxima and minima appgars to have been proposed for a prize exercise in the University of Leyden in 1823. Essays by Verdam and Verhulst obtained prizes ; they were published in 1824. The title of the two essays is the same . . . Commentatio ad Qucestionem Mathematicam ... in Academia Lugduno-Batava . . . propositam . . . Verdam's essay occupies 100 quarto pages ; from page 76 to the end is devoted to the Calculus of Variations. The writer confesses that he has a very imperfect knowledge of this branch of the subject. Some of the ordinary formulae are given, but the demonstrations are only sketched, and reference is made to La- croix for the details ; a few of the usual problems are given in illustration. The essay is not free from error ; we may refer for example to the treatment of the limiting equations. Verdam says in effect, that in a term of the form ABt/, if the limits of i/ are fixed we still have A = 0, whereas the term vanishes because Sz/ = and the relation -4 = does not in general hold. And on 442 . VEEHULST. AIRY. BORDONI. his page 94 he gives an example from Euler's Methodus Inveni- endi ... page 88, which he treats hj means of that formula given bj Lacroix which we have discussed in Art. 38. Verdam's re- sult is correct for the case which Euler considers in which L is a function of 11, but is not true if, as Yerdam says, Z/ is a function of x and y. _ Yerhulst's essay occupies 30 quarto pages ; about three pages are devoted to the formulae of the Calculus of Variations, and three more to some of the common problems. 377. Yerhulst. There is another essay by Verhulst, which is on the Calculus of Variations exclusively. This obtained a prize which was offered in 1823 by the University of Ghent, and was published in 1824 under the, title . . . Commentatio ad Quces^ tionem Mathematicam ... Academice Gandavensis ^ropositam.... The essay contains a brief sketch of the subject, and discusses seven problems ; it gives some account of the application of the subject to Mechanics, and demonstrates the principle of least action. It is chiefly remarkable for grave errors. 378. Airy. In Airy's Mathematical Tracts, published at Cam- bridge in 1826, twenty-three pages are devoted to the Calculus of Variations. These pages form an excellent elementary treatise on the subject. The author in his preface speaks of the subject as the "most beautiful of all the branches of the Differential Calculus." He says of his treatise, " by adhering rigorously to principles, by exemplifying every formula, and by avoiding the investigation of useless theorems, the author hopes that he has removed many of the difficulties which have been thought to beset this theory." The fourth edition of the Mathematical Tracts was published in 1858 ; the treatise on the Calculus of Variations is here increased by two pages, namely pages 240 and 241 of the work. 379. Bordoni. Lezioni di Calcolo Sublime. Milan, 1831. This work is in two octavo volumes ; the Calculus of Variations BOEDONI. 443 occupies pages 192 — 298 of the second volume. Bordoni adopts the method and notation of Lagrange which we have described in Art. 15 ; and the work is rendered extremely perplexing bj the profusion of dots and dashes and affixes with which the symbols are loaded. Scarcely any examples are given in illustration of the theory. This appears to be the first elementary work which introduced Poisson's formula for the variations of the differen- tial coefficients of a function of two independent variables ; see Art. 262. We will notice a few points in the treatise in detail. 380. Two examples of the use of Variations are given by Bordoni on his pages 261 — 265, which we will briefly explain. I. Suppose a fixed surface and two fixed points outside it ; let a string have its extremities fixed to these points, and let it be stretched and kept in contact with the surface by means of a point moving on -the surface and against the string ; thus the whole string consists of four portions, namely two straight lines outside the sm-face and two curved portions on the surface. The moving point will trace out a locus on the sm-face after the manner in which an ellipse is traced out on a plane by a moving point which stretches a string having its ends fixed. Then the locus traced on the sm-face has this property analogous to a property of the ellipse ; the tangent at any point of the locus makes equal angles with the two curved portions of the string- meeting in that point. This we shall now prove. The string is inextensible, and therefore the sum of the variations of the four parts is zero. Let x, y, z denote the co-ordinates of a point in one of the curved portions, so that the length of this portion is \\l{^-\-y''^-\-z'^)dx between proper limits, where y stands for -j- and z stands for -y- . The variation of this integral according to the usual notation consists of an integrated part 444 BOEDONI. and an unintegrated part Now this unintegrated part vanishes, because we know from statical considerations, that the curved portions of the string assume the forms of the lines of maximum or minimum length on the surface, and for such lines the unintegrated part of the variation of the length of an arc vanishes. We have therefore only the integTated part remaining, and this may be put in the form Bx + y'hy + zBz that is, Zs cos ^, where Ss^ = B^ + By^ + Ss^, and ^ is the angle between two lines, one having its direction-cosines proportional to Sec, By, Bz respectively, and the other having its direction-cosines proportional to 1, y\ z respectively. Now at the point which is common to one of the straight portions of the string and one of the cm-ved portions, Bs and ^ have the same values for each portion ; so that the two terms which are thus contributed to the variation of the whole length cancel. Then at the point common to the curved portions Bs is the same for the two portions, and therefore ^ must have the same value in order that the whole variation may vanish. II. Suppose one end of a string fixed to a point in a curve on a fixed surface ; and let the string be stretched so that a part is kept in contact with this curve, a part kept in contact with the sm-face, and a part is free from the surface. Then whatever may be the position of the string, provided that the three parts are kept stretched, the third part is always a normal to the surface traced out by its free end. This is proved in the same manner as before. Let ^, 17, f be the co-ordinates of the free end ; x, y^z the co-ordinates of the point where the string leaves the surface. Let s^ be the length of the straight portion, s^ the length of the part which is only kept BORDONI. 445 on the surface, 53 the length of the part which is kept against the curve. The whole variation of s^ + s^ + s^ must be zero. The unintegrated part of the variation of s^ vanishes as before; the only variation in s^ is that which is produced by lengthening or shortening the portion in contact with the curve ; and this variation is cancelled by the corresponding term in the integrated part of the variation of s^. The variation of s^ so far as it depends on the variation of the point {x, y, z) is cancelled by the corre- sponding term in the integrated part of the variation of s^. Thus that part of the variation of s^ which arises from the variation of the point (|, 7;, t) must separately vanish. But sl = [x-^Y+{y-'nT+{z-^)\ therefore (^- ^) ^H (3/-.) ^^ + (^-Q ^^^p. this equation shews that two lines are at right angles, namely the line which has its direction-cosines proportional to h^, Stj, 8^ respectively, and the line which has its direction-cosines pro- portional to X — ^, y — 7), z — ^ respectively. This proves the theorem. 381. On pages 281 — 298 of his work, Bordoni discusses the criteria for distinguishing between maxima and minima values ; here he follows the method of Legendre. Suppose we have to investigate the maximum or minimum of I ^ {x, y, y') dx. The terms of the first order are supposed treated in the usual way We have then to examine the sign of j{A{ByY + 2BByBy' + G{By'Y}dx, . . d'<\> ^ d' ^ d' where A = -^, S = ^^,, C=^,. Now we have identically, whatever a may be, A{Byy + 2BSyBy'+C{Byr = {A-a'){ByY + 2{B-o)ByBy'+C{Byr+{a{Byy}'. 446 BORDONI. Thus the above integral "becomes a{B^Y + j{{A-a') {hyf +2{B-a)hy^y' -^ C [ZyJ] dx. Then if a can be found so as to make {A — a!) G greater than {B — af, the sign of the expression remaining under the integral sign will be the same as the sign of (7; and thus we shall be able to determine whether there is a maximum or a minimum. A suitable value of a may be found thus. Let c be the least value of G between the limits of integration, h the least value of A—B'; find yu. from the equation SO that IX, = V(Jc) ^^ , 1 - ^e Vc where ^ is a constant. Then a=B—fi is a suitable value. For h + iJb is less than A— B' + [jJ, and c less than G ; therefore {h+/jb')c is less than G{A^B' + fi), that is G{A — B' + jx) is greater than jj?, that is G {A^ a) greater than {B — a)^. Bordoni does not however allow for the exceptions which may arise ; thus in applying the test to a geodesic line, he says that such a line is a line of minimum length, which we know is not necessarily the case. 382. One of the investigations which Bordoni gives is in- tended to discriminate between the maximum and minimum of jcf) {x, y, z, y', z) dx when a relation F {x^ y^ z, y\ z) =0 is sup- posed to hold. In this case by the use of a multiplier X we find that we have to investigate the sign of the terms of the second BOEDONI. 447 order in the variation of I (<^ + \F) dx. These terms form a poly- nomial of the second degree in hz, S?/, Ss, %; then hz is eliminated by means of the relation We thus obtain under the integral sign a polynomial of the second degree in hy, hz, By, say A{Bijy + 2BS2/S2j'+ + ^%T; - , ^ d^^ + XF)- ^ d'{cj> + \F) d^ {s) dx is to be a maximum or minimum while I V(l +p') dx is constant, we proceed to find tlie maximum or minimum of I \i>{s)-^\ '^{).-\-jf)\dx. We can tlien apply the formula of Art. 38, supposing that The integrated part of 8 I Vdx in that formula vanishes because the limiting values of x and y are constant ; hence for a maximum or minimum we have P+ (^ - 7) P' = a constant = Q say, therefore Xp + (^ - 7) ^ = (7 V (1 +/ ) , (1); therefore ^ + ^-^=|v(i+P^) by differentiating we obtain dl -G dp dx fs/{\+f)dx' that is L therefore .,,. ds C dp "^^'^d^-^fd^ therefore ^(,) = _|+c;, therefore dy _ G dx G,-{s)f]' MOMSEN. 455 From the last two equations x and y must be found in terms of 5 ; two constants already appear, and two more will occur in the integrations for finding x and y. These constants must be de- termined from the consideration that when 5 = the values of X and y must be the co-ordinates of the given point A, and when s is equal to the given length the values of x and y must be the co-ordinates of the given point B. A different view of this part of the solution is given in De Morgan's Differential Calculus^ page- 468. ,In the particular case considered bj Momsen <}){s) = s', thus ~ = -pz : and this shews that the curve must be a catenary dx C^-s' '' having its directrix parallel to the axis of y. And thus we see that the ordinary figure with A and B on the axis of x is impossible in the present case, because a catenary cannot be cut in two points by a straight line perpendicular to its dhectrix. We proceed to investigate the terms of the second order in the variation of / -j^ (5) + X ^(1 +i?^) r ^^ iii the particular case in which ^ [s] = s. We have to the second order . Hence by the nature of the catenary it may be shewn that X is numerically equal to the distance of the point B from the directrix of the catenary. Suppose in the first place that the ordinate of A is less than that of J5 ; if the catenary is concave to the axis of x, then \ is negative and is numerically greater than a, so that \+ a —x\^ negative and we have a maximum ; if the catenary is convex to the axis of x, then \ is positive and we have a minimum. Next suppose that the ordinate of A is greater than that of B ; if the catenary is concave to the axis of x, then \ is positive and we have a minimum ; if the catenary is convex to the axis of x, then X is negative and is nume- rically greater than a, so that X + a — a; is negative and we have a maximum. The last eight lines of Momsen's investigation are unsatisfac- tory ; he comes to the conclusion that there is always a minimum. He argues thus ; let W=\\s + \^J{\+f)]dx^ Jo by integration by parts we have I sdx = sx—\ X -J- dx, •'0 •' 7"^ therefore I sdx= j {a — x) -j- dx ; , if MOMSEN. 457 thus W=r{\ + a-x)*/{l+p')dx. •' Hence Momsen sajs that W is of the same sign as the ex- pression of the second order (\ + a — a;) {BpY dx ^ and this is true from what we have given although Momsen does not prove it. Then Momsen concludes that the result necessarily makes W a minimum. This is inadmissible ; the sign of TFhas nothing to do with the question of the maximum or minimum of I s dx. J It should he observed that the solution here given is liable to fail, for the given length of curve may be too great to constitute an arc of a catenary joining the two given points. We will consider one case, and treat it after the manner of Art. 352. Let k^ and ^•^ be the ordinates of ^ and J5, and suppose k^ less than \. Suppose a maximum is required, and let us try if the problem can be solved by supposing the curve joining A and B to be made up of a straight line of length y^ — ^ formed by producing OA through A, and an arc joining the point (0, yj to B. The expression which is now to be a maximum is /• a rx I {t/q — kg + s] dx, where s = I V'(l +P^) ^^ \ Jo Jo and the whole length is ^o ~ ^o+ / V(l +i?^) dx. •I Thus we may consider that we have now to find the maximum of •'0 that is of I (s + \ V(l +i>') ]dx+ia + \) {y^ -h^. The only point in which the solution will differ from that formerly given is in the terms outside the integral sign. We have (a + \) hj^ from the term {a + X) 3/0 ; and there is the term 458 MOMSEN. {P+ AF—IP') CO in the notation of Art. SSjVhicli gives us — CSi/^. Thus we require that a + \-C = (5). We have already stated that from (4) it follows that \ is nume- rically equal to the distance of £ from the directrix of the catenary ; then from (3) it follows that G is numerically equal to the para- meter of the catenary, that is, to the distance of the directrix from the nearest point of the curve. And if the catenary is concave to the axis of x both \ and C are negative. Thus we shall deduce from (5) that the catenary must toitch the axis of ^ at the point (0, %)' This holds so long as y^ is not greater than Jc^. If we cannot consistently with the given length have y^ not greater than \, we must make the catenary convex to the axis of x and make it touch the axis of y at the point (0, yj. If a minimum be required, the curve consists of a catenary beginning at A and ending at the point (a, ^J, and having its tangent parallel to the axis of y at this point ; and the length con- sists of that of the arc of the catenary together with that of the line joining the points (a, yj and (a, \). 388. The following problems relating to the maxima and minima values of double integrals are solved in Momsen's fourth section, the limits of x and y being supposed given in all cases. (1) The maximum of llzlx^ + y^—azjdxdT/; this is in Strauch, Vol. ii. page 562. (2) The maximum of [jk V {x^ + f + z^)--\dx dy, (3) The minimum of [fj^ V(«'' + 2/') + | (5; - cA dx dy. (4) The surface of maximum or minimum area having a given boundary. (5) The surface of maximum or minimum area among all those which correspond to a constant volume. MOMSEN. 459 (6) The volume of a solid being given, it is required to find its bounding surface so that its centre of gravity may be at a maxi- mum distance from the plane of {x, y). This problem is in Strauch, Vol. II. page 610. The problem is analogous to that considered in Art. 340 ; the result is that the required surface is a plane. Both Momsen and Strauch encumber their solution by not paying atten- tion to the remark at the end of Art. 340. (7) Among surfaces of given area to find that which has its centre of gravity at a maximum distance from the plane of {x, y) ; the difierential equation of the required surface is here obtained, and as in the preceding problem the investigation is needlessly encumbered. A general formula for the variation of double integrals is given by Momsen from Lacroix, which involves the errors already indi- cated ; see Art. 27. 389. Besides the errors we have already noted in Momsen's treatise, a few more may be given. On his page 32 Momsen is speaking of the determination of the constants in the problem we have given in Art. 65. He has a con- dition equivalent to \j, = 0, so that ;7- (-2) must =0 when x =h. This equation which holds for a particular value of x he integrates, and deduces z = Cp^, which is inadmissible. On his page 39 Momsen is speaking of the determination of the four constants which occur in the solution of the problem of the brachistochrone in a resisting medium. He says that two are to be determined by making the curve pass through given initial and final points; he proposes what he considers two conditions for determining the other two, but these two conditions amount really to only one condition. The condition which he omits is that the initial velocity must be supposed given, as he has really assumed at the top of his page 38. Remarks similar to that which we have noticed in Art. 387 as occurring in the last eight lines of Mom- sen's investigation occur in other places of Momsen's treatise ; see his sections 36, 37, and 40. At the end of his section 48 he 460 MOMSEN. ABBATT. DE MOEGAN. assumes without any proof that the sign of C can Ibe easily ascer- tained to be positive when the curve is convex to the axis of x ; the quantity G is the same as that denoted by M in Strauch, Vol. II. page 516, and its sign is not determined by Strauch. In his sections 38 and 58 Momsen retains terms which are absolutely zero, in the same way as i hy dx is zero in Art. 340. 390. Abbatt. A Treatise on the Calculus of Variations, by Kichard Abbatt. London, 1837. This is a volume in foolscap octavo, of 207 pages, with a preface of 1 1 pages. The writer in his preface refers to Lacroix, Lagrange, Euler, Woodhouse and Airy ; and on page 203 he refers to Pois- son's memoir. He appears to have used Poisson's memoir also on his pages 18, 62, 115 — 121 and 194 — 203. Nevertheless he gives on his pages 192 and 193 the erroneous formulas which we have noticed in Arts. 39 and 40. He gives the correct formulee on his pages 197 and 198, but his mode of obtaining them is not satis- factory. The treatise contains numerous examples selected from preced- ing writers on the subject. 391. De Morgan. In Professor De Morgan's Differential and Integral Calculus, pages 446 — 475 are devoted to the Calculus of Variations ; this part of the work was published in 1840. By adopting a condensed yet expressive notation a large quantity of information on the subject is compressed into a brief space. There is no investigation of the terms of the second order, but with this exception the student is introduced to all the important parts of the subject. In the formulee respecting the variation of double inte- grals the limits with respect to both variables must be understood to be all constants, for the reason which we have given in Art. 28. On pages 470 and 471 the problem of the brachistochrone in a resisting medium is discussed, and the way of determining the fom- constants which occur is carefully explained. Mr De Morgan observes that this part of the problem is omitted in silence by Woodhouse and Lacroix, and that " Lagrange merely says that Bz^ is indeterminate, but does not give any reason..." Lagrange's DE MOEQAN. COURNOT. HALL. BRUUN. 461 meaning is to be found however from wliat lie had previously given on page 465 of the Legons ..., edition of 1806 ; and it appears from this that Lagrange's view was correct. We may observe that in Stegmann's discussion of the problem the constants are determined in the same way as by Mr De Morgan ; see Stegmann's work, pages 318 — 321. Strauch is not satisfactory on this point ; see his Vol. ll. page 418. 392. Coumot. Two chapters are devoted to the Calculus of Variations in Cournot's Traiti. eUmentaire de la Theorie des Fonc- tions... Paris, 1841. These chapters occupy pages 113 — 155 of the second volume of the work ; they form a good elementary treatise on the subject. It should be observed however that in the varia- tion of double integrals Cournot reproduces the error which we have explained in Art. 27. 393. Hall. The Encyclo^cedia Metropolitana contains a brief treatise on the Calculus of Variations by Professor Hall. It occupies pages 209 — 226 of the second volume of the first divi- sion of the Encyclopaedia; the date of this volume is 1843. The treatise gives the usual theory so far as terms of the first order in the variation of single integrals, and applies the theory to a few examples. 394. Bruun. A Manual of the Calculus of Variations. Odessa, 1848. This is an octavo volume of 195 pages in the Eussian language. The difficulty of the language will prevent any detailed account of the work. It is divided into four parts. The first part occupies pages 1 — 36, and gives the variations of expressions. The second part occupies pages 37 — 56, and discusses the criteria of integrability of expressions. The third part occupies pages 57 — 181, and con- tains the investigation of maxima and minima values. The fourth part occupies pages 182 — 195, and consists of a sketch of the history of the subject. In the third part of the work the terms of the second order in the variations of integrals are investigated with the view of dis- tinguishing between maxima and minima values. Bruun takes in 462 BRUUN. succession the case in wliich the function under the integral sign involves only a?, y and y\ and the case in which the function under the integral sign involves a?, y, y and y" , He gives hoth Le- gendre's method and Jacobi's method, and of course by comparing the results of the two methods the. auxiliary quantities introduced by Legendre's method become determined ; see Art. 235. The only passage in the third part which presents any appear- ance of novelty is that on pages 103 — 108. After having finished the discussion of the method of Jacobi applied to I ^ (oj, y^ y, y") dx, Bruun intimates that this method is very complex, and that he will explain another method of discriminating between maxima and minima values, given by Sokoloff. He does not however do more than introduce the method to the reader and refer for detail and ex- emplification to the memoir of Sokolofi*. The title of this memoir appears to be Researches on a certain point of the Calculus of Varia- tions. Charkoff, 1842. The following process will give an idea of the method so far as it is explained by Bruun. Suppose we are investigating the sign of the terms of the second order in the variation of | j> {x, y, y') dx. The expression we have J Xq to examine may be written thus, \'^{A[hyY + 2B^yhy'+G{hy'r]dx, where A, B, G are functions of x. We wish to know if this ex- pression retains the same sign for all values of Sy and By' ; we will test this by ascribing a certain convenient value to Sy. We have [{A {Byf + 2B Sy Sy' + C {Szj'Y} dx = j{{ASy + BSy') Sy + {B Sy + C Sy') Sy'} dx = {BSy+ CSy') Sy+j{ASy+BSy'--^iBSy+ CSy')\ Sy dx. Thus if we choose for Sy a value which makes d_ dx ASy + BSy'-i-{BSy+CSy')=0, BRUUN. PRICE. " 463 the term we wish to examine can be actually integrated, and so its sign can be .easily ascertained. Now a value of Sy which will satisfy the above equation can be found, supposing that we can solve the differential equation which arises from making the terms of the first order vanish in the variation of I ^ {x, y, ?/') dx ; see Art. 251. Such a value will be of the form ^^u^-^-^^u^, where ^^ and ySg are arbitrary constants, and u^ and u^ are known functions of X. Thus {BSt/ + C8i/') hy will be a homogeneous function of the second order of the arbitrary constants ^^ and ^^, and so we may by ordinary methods investigate whether {{BZy+ Chy') hy],- {{Bhy + C^y') hy\, is positive or negative for all values of these arbitrary constants. Such appears to be essentially all that Bruun gives. It is 'obvious that by this method we may in some cases succeed in shewing that a proposed expression has neither a maximum nor a minimum value ; but it does not appear obvious how we can deduce a positive test which shall shew when a proposed expression is a maximum or a minimum. The pages 193 — 195 of the work contain a list of references to writers on the subject. In this list, besides the memoir of Sokoloff, two works are named which the present writer has not had an opportunity of consulting. These works are the following. Textor. Kurze Darstellung der hohern Analysis, nehst einem Anhange von dem Variationencalcul. Berlin, 1809. Senff. Elementa Calculi Variationum. Dorpat, 1838. The present writer is indebted to the kindness of Professor Bruun for a copy of his Manual of the Calculus of Variations. 395. Price. A treatise on the Calculus of Variations forms part of the second volume of Professor Price's Treatise on Infinitesi- mal Calculus. Oxford, 1854. The Calculus of Variations occupies pages 234 — 334 ; and there is an application of the subject to the conditions of integrability on pages 440 — 446. The author refers to the works of Euler, Lagrange, Poisson, Jacobi, Ostrogradsky, 464 PEiCE. Delaunay, Strauch, Jellett, and Scliellbacli. The treatise gives the usual theory of the variation of single integrals ; it explains Jacobi's method of distinguishing between maxima and minima values ; it treats copiously of geodesic lines ; and it touches briefly on the variation of double integrals. 396. It may be of service to a student of Professor Price's work to refer to a few points in which he may find some dif- ficulty. In Art. 93 we have given Poisson's proof of a certain relation, namely, Hy + Kz = 0, and we have stated in Art. 94, that La- grange had proved this relation repeatedly. Mr Price on his pages 269 and 272 makes a remark which amounts to assuming that this result is obvious without demonstration. On page 270 some remarks are made on the method of deter- mining the arbitrary constants which occur in solving problems in the Calculus of Variations. If the limiting values of the quantities are not restricted, the coefiicients of the terms ^x^, Sa?^, hy^, By^, ... must be equated to zero. The book proceeds, " Suppose, however, that equations are given connecting the variables at the limits, that is, that equations are given between cc^ and y^ and between x^ and y^ : then if T = is the integral of H = 0, there will be given „ fdT\ (d^\ f d'^^-'T ^ fdT\ (d'^'-'T °' uy): \df): "• W-'k '' \dyk '" W ' This seems unsatisfactory. If, for example, T = then T^ = and T^ = necessarily, and no new information is supplied by these equations. The true method when relations are given between the limiting values of quantities is to deduce relations between the vari- ations of these limiting values ; thus some of the variations are expressible in terms of the others, and the number left arbitrary is diminished; we then equate to zero the coefficients of these re- maining variations, and the equations so obtained together with the given relations can be used to determine the arbitrary constants. In some cases, as we have seen, in Arts. 276 and 367 the number of arbitrary constants occurring in a solution may be too small. Mr Price speaks of such a problem as indeterminate on PRICE. 465 page 270 ; it should rather he called impossthle, for the prohlem cannot be solved at all, unless certain restrictions are imposed. See Mr Jellett's Treatise, page 44. On page 274 instead of " ... F^ (a, b) =0, F^ (a, h) = 0. By means of which four equations we can determine -^ , a and b, and thereby definitely fix the line whose equation is (32)," read -...F,{x„i/,)=0, F,{x„i/,)=0; also^i^ = ^^. We have now five equations for finding x^, y^, a^is y^ 3 •" An important mistake occurs on page 296. The equations (131) cannot be deduced in the way given in the book. Equations (131) involve important properties of geodesic lines, but the equations (127), (128), (129), (130), from which the book deduces (131), are not at all restricted to geodesic lines. Equations (131) may be proved thus ; we may shew by direct investigation that dk dfjb dv ds d^x — dx dh ds d'^y —dyd^s ds d^z — dz dh ' and then by means of equations (115) of the book, we have dx _diM _ dv 'U~T~W" On page 307 we read " suppose a series of geodesic lines to originate at a point (/x^, v^) and to touch the line of curvature (fi^) ; then at that point...." It is not possible to have a series of geodesic lines passing through a point and touching a given line of curvature. In fact the words "originating at a point" should be omitted, as they are not required or used. On page 308 we read " it may be proved in the same way as the analogous theorem in plane geometry, that the geodesic radii vectores make equal angles with the curve of curvature." The proposition in question has been assumed to be true in writing the equations dr^ = dyS cos i, dr^ — — dyS cos «*, 30 466 PEICE. METEE. wliicli occur immediatelj before, so tliat of course we cannot use any consequence drawn from these equations in order to establisli the proposition. On page 329 the same mistake occurs which we have noticed in Art. 232. We have an equation /, /s^'^-^.©'+s^»(i and it is stated that any value of u which makes hH = will also satisfy the right-hand member of the equation. Either the limits and 1 should be omitted from the left-hand side and then the conclusion is that any value of u which makes hH= will make the right-hand member equal to a constant ; or if the limits and 1 are retained on the left-hand side the right-hand side must be written {^-(i)'+s^^(i)"}.-{^.(~)'+l^.©l- and then any value of u which makes hH= makes this expression vanish. 397. There are some passages in the treatise which do not appear treated with sufficient detail for those who are studying the subject for the first time. For example, the process of page 258 of the treatise may be compared with Ostrogradsky's corresponding process, which we have reproduced in Art. 128. The statement on page 283 respecting the equating certain ratios to a constant quantity seems to need explanation. On page 310 it is stated that the directions of the principal lines of curvature at any point of an ellipsoid are evidently parallel to the principal axes of a section of the ellipsoid made by a plane parallel to the tangent plane at the point in question ; they are parallel but not evidently so without demonstration. 398. Meyer. Nouveaux elements du Calcul des Variations. Liege et Leipzig, 1856. This treatise consists of 132 octavo pages. In the preface the author says he has preserved the classification of variations into simple and compound, pure and mixed, given by Strauch, and that METEE. 467 he has borrowed from the profound work of that writer the formula for the variation of a double integral when the limits of the first integration are themselves susceptible of variation. He says that he has given a new method of explaining the principles of the subject, and he considers this method to have the threefold advan- tage of deducing the subject from Taylor's Theorem, of freeing it from the consideration of infinitesimals, and of freeing it from any question about the convergence of series. As he only proposed to write an elementary treatise he has not entered upon the calculation of the variations of the second order. He says that for isoperime- trical problems he has given a method which is substantially Euler's, but that he has introduced a modification which removes some objections that are brought against the methods of Euler and Lagrange. Eor the composition of the treatise he has consulted the most eminent writers, especially Euler, Lagrange, Poisson, Dirksen, Ohm and Strauch ; but as his method of explaining the principles of the subject differs from those of all the authors whom he consulted, he calls his treatise, New Elements of the Calculus of Variations. The treatise however does not seem to possess any claims to attention ; the method which the author adopts for explaining the principles of the subject would probably present serious difficulties to a beginner. In many of his earlier formulee Meyer retains terms of a higher order than the first ; this is a useless encum- brance, because he makes no use of those terms afterwards. It should be added that the book has been obviously printed at a press which is rarely used for mathematical works, and thus it presents an awkward and almost repulsive appearance. Meyer's method will be seen from the following example which he gives. JjQtfix) and F{x) be two functions of cc; form the equation f{x + ri)=F{x)', from this we may find for rj a value, say r) = ^(x), so that iden- tically f[x + ^{x)]=F{x). Thus the original function f{x) changes its properties and is transformed into F{x). For example, put a {x -\- 7}) = ska. x, 30—2 468 MEYER. sin X—- ax then 7) = a ( sma; — aajA and a\x-\ ) = sm x. Thus if y =f{x), Meyer puts Dy = F[x) -fix) =/(a.+ 77) -/{x) Thus with him By = -^7j, S^y = -r^ ^7^ ..., where 77 is an arbi- trary function of x. This method appears unsatisfactory. In the first place it is deficient in generality. The usual method is to suppose f{x) changed into <}> {x, t) and not necessarily into the restricted form f[x + t). Meyer seems to want to consider Zy and Z^y as arbitrary and unconnected J this however is not the case in his system, for \dxj In the next place, a beginner would be perplexed by the author's speaking of i; as a constant, after the explanation and example which have been given of it. This language occurs how- ever on page 6. Again, on page 22 we have this process. . Having d"^'u . . « . given the fimction p = -5-^ , in which x is the constant element, required Sp, 6^, ..., y being a fimction of x. METER. 469 We have by definition ~ dx"^^ dx"^ "^2 dx"^ ^"'' but sy=|^, %=g^^^...,• moreover rj being an arbitrary function of the constant element x, we must regard ?;, rf, .., as constants.... Here the last statement would appear obscure to a beginner. On page 81 there is some novelty, but it cannot be com- mended. The subject of isoperimetrical problems is considered on pages 89 and 90; but it is not obvious what modification or improvement the author has made of the common method. CHAPTEE XV. MISCELLANEOUS AETICLES. 399. The present chapter contains a brief account of some miscellaneous articles connected with the Calculus of Variations ; the connection is in some cases very slight, hut it is useful for purposes of reference to notice all the articles which bear on the subject. The notices will take the articles in chronological order. 400. Ampere. Remarks on the application of the general for- mulse of the Calculus of Variations to mechanical problems. This memoir was published in 1805, in the first volume of the Memoires presentes a V Institut ... par Divers Savans. It occupies pages 493 — 523 of the volume. This memoir contributes nothing to the theory of the Calculus of Variations ; its only interest arises from its relation to mechanics. Lagrange had remarked in the Mecanigue Analytique, that there is an analogy between the equa- tions of equilibrium in mechanical problems and the equations furnished by the Calculus of Variations for determining the maxima and minima values of integral expressions. Ampere makes some general remarks on this analogy ; he illustrates his remarks by the example of a uniform inextensible string suspended by its extre- mities and acted on by gravity. In connection with this example he indicates several properties of the common catenary. On his page 503 Ampere makes some remarks to the following effect. The Calculus of Variations consists of two parts, one in which it is sufficient to attribute variations to the dependent vari- ables only, the other in which variations must be attributed to all the variables dependent and independent; writers on the subject AMPJlEE. LAGEANGE. CEELLE. GEEGONNE. 471 have however confined themselves as much as possible to the former part. The theory of the Calculus of Variations is therefore, according to Ampere, not yet established upon absolutely rigorous principles. There remains in this respect a deficiency in Mathe- matics which Ampere proposes to consider elsewhere. It does not however appear that this purpose was accomplished ; for the memoir in Gergonne's -4 mia/e.? ... which we have noticed in Art. 375, can hardly be considered of sufficient importance to correspond to the purpose here expressed. On his page 516 Ampere refers to some other memoir, without however indicating where it is to be found. 401. Lagrange. The first volume of the second edition of Lagrange's Mecanique Analytique was published in 1811; the second volume was published in 1815, after Lagrange's death. Lagrange uses the notation and the processes of the Calculus of Variations freely throughout the work, but the great interest which belongs to his investigations is derived from their connection with Mechanics. The theory of the Calculus of Variations receives no accession from the work. 402. Crelle. In the article Variationsrechnung of Kliigel's Matheniatisclies Worterhuch, page 713, reference is made to a work by Crelle. The article says, " Crelle' s views on the principles of the Calculus of Variations seem not sufficiently known ; they are contained in his Versuch einer rein algebraischen Darstellung der Rechnung mit verandlichen Orossen, I. Gottingen, 1813, pages 527 — 776. The numerous new symbols render the work difficult for study. The application to maxima and minima is not included in the work." The present writer has not seen this work by Crelle. 403. On the surface of minimum area between given limits. Gergonne's Annales de Mathematiques, Vol. 7, pages 68, 99, 143—156, 283—287. 1816. These pages contain some problems proposed for solution ; the problems are particular cases of the question of the surface of 472 GERGONNE. minimum area, various conditions being given with respect to tte limits of the required surface. There is an attempt at the solution of one of the problems by M. Ted^nat, and criticisms on this attempt by Gergonne; Gergonne also makes some observations on the general question. The particular case considered by Tdd^nat is the following ; it is required to determine a surface which shall pass through the inverse diagonals of two opposite faces of a cube, and so that the area of the portion of the surface intercepted by the cube may be a minimum. By mechanical considerations relating to a flexible elastic membrane, Teddnat considers that he proves that the surface TTZ must be that which is determined by the equation y = x tan — . Gergonne admits that this surface satisfies the general partial differential equation for a surface of minimum area, but objects that it is not proved that this surface gives the solution of the problem with the prescribed limiting conditions. Gergonne's criticisms indicate that he had considered the problem more closely than Tedenat had. Gergonne gives an interesting account of the circumstances which drew his attention to these problems. A distinguished mathematician informed Gergonne that he had serious doubts as to the legitimacy of the methods given in the Calculus of Variations. Gergonne invited him to write an article upon the subject which might appear in the Annales . . . ; but the article was never sent for publication. One of the objections of the distinguished mathe- matician is expressed thus ; suppose the so-called minimum surface to be determined by conditions which preclude it from being a plane surface ; draw any plane curve upon it ; then remove the piece of the so-called minimum surface which is bounded by this plane curve, and replace it by a plane having the same boundary ; thus a surface is obtained which is less than the so-called minimum surface. Gergonne replies that this objection only amounts to a proof that it would be impossible to draw on the minimum surface a plane closed curve ; and this impossibility is consistent with the fact that the minimum surface has at every point its principal curvatures in opposite directions. GEEGONNE. CEELLE. 473 Gergonne states that it appeared to him that it would be useful to propose certain problems relative to the minimum surface in which there should be definite limiting conditions. Besides the problem already given, the following are proposed. To find the surface of minimum area among all those which are bounded hy the curve of intersection of two cylinders of the same radius, the cylinders having their axes at right angles to each other, and the axis of each cylinder being a tangent to the other cylinder. A quadrilateral is given having its sides not all in the same plane; find the surface of minimum area among all those which are bounded by the sides of this quadrilateral. Find the surface of minimum area among all those which are bounded by two circles given in magnitude and position. Find the surface of minimum area among all those which are bounded by the sides of a given square, and which include between themselves and the square a given volume. Among all surfaces which are bounded by the sides of a given square, and which have within this boundary a given area, find that which includes between itself and the square a maximum volume. ' No attempts seem to have been made to solve these problems, except that Ted^nat intimates that he believes that no continuous surface can be found as a solution of a certain special case of the problem in which the given boundary is a quadrilateral having its sides not all in the same plane ; see page 286 of the seventh volume of the Annates. ... 404. Crelle. Remarks on the Calculus of Variations. These remarks form part of a collection of mathematical treatises published by Crelle under the title of Sammlung MathematiscTier Aufsdtze und Bemerhungen. The work consists of two octavo volumes; the first was published in 1821, and the second in 1822, both at Berlin. The remarks on the Calculus of Variations occur in the second volume ; they occupy pages 44 — 174. These remarks constitute an elementary treatise on the subject ; the treatise however does not seem to possess any special merit, and the 474 CEELLE. MULLEE. BOOLE. notation is repulsive. On liis page 47 Crelle refers to his former work on the subject, but not in terms of commendation; see Art. 383. 405. Crelle. Eemarks on the principles of the Calculus of Variations. . This memoir forms part of the Transactions of the Academy of Sciences of Berlin for 1833 ; the date of publication of the volume is 1835. The memoir occupies 40 pages ; it proves the ordinary formula3 for the variation of a single integral, both for constant and variable limits of integration. The method and notation differ from those in common use, but present no obvious advantages. 406. Miiller. On establishing and extending the Calculus of Variations. Crelle's Mathematical Journal, Vol. 13, pages 240 — 249. 1835. This article contains some general remarks on functions without any obvious reference to the Calculus of Variations. At the end of the article the author says that he will on another occasion explain the method of applying these remarks ; it does not however appear that this design was accomplished. 407. Boole. On certain theorems in the Calculus of Vari- ations, Cambridge Mathematical Journal, Vol. 2, pages 97 — 102. 1840. The author says at the beginning of this article, " It would perhaps have been more just to entitle this communication ' Notes on Lagrange.' The papers from which it is selected were written towards the close of the year 1838, during the perusal of the Mecanigue Analytique^ The article contains a simple demonstra- tion of a theorem which forms the basis of Lagrange's investigations on the great problem of the variation of the arbitrary constants. The theorem is that which Mr De Morgan speaks of as "perhaps the most characteristic specimen of the genius of Lagrange which could be given;" see his Differential and Integral Calculus, page 532. The author thus indicates the object of the latter part of his article. " I shall now proceed to demonstrate from the general BOOLE; DELAUNAT. STRAUCH. 475 transformed equation of motion the principles of the conservation of living forces, and of least action. The former of these has been thence deduced by Lagrange. I am not however aware that the latter has been obtained from the same equation, either bj the discoverer of the Calculus of Variations, or by any subsequent author." 408. Delaunay. On the surface of revolution which has its mean curvature constant. Liouville's Journal of Mathematics, Vol. 6, pages 309—315. 1841. "When we investigate the problem of finding the surface which with a given area includes a maximum volume, we arrive at a certain partial difierential equation which expresses that the sum of the principal curvatures at any point of the surface is constant. Delaunay proposes to determine what surface of revolution has this property. He finds that the generating curve must be such as would be traced out by the focus of a conic section, if the conic section itself were to roll without sliding on a fixed straight line. There is a note by Sturm immediately after Delaunay's article, in which the same result is obtained in a different manner. The result is also given in Mr Jellett's treatise; see his page 364. 409. Strauch. Problems in the Calculus of Variations. Grunert's ArcMv der Mathematik und Physik, Vol. 3, pages 119—195. 1843. This article contains some problems which Strauch published as a specimen of his work on the Calculus of Variations. The first seven pages of the article contain some introductory remarks and definitions, and then follow the problems. A few of the problems relate to expressions involving neither symbols of differ- entiation nor symbols of integration ; the remainder relate to ex- pressions which involve differential coefficients but not integrals. All these problems are reproduced by Strauch in his work. In the same volume of Grunert's ArcMv ... a few remarks are made on Strauch's article by Gopel ; these remarks occupy pages 405—407 of the volume. Gopel says that the problems of the first 476 STRAUCH. LAURENT. kind whicli Strauch considers are only ordinary problems of maxima and minima values; and he makes a few other observations. Gopel's remarks did not convince Strauch of the necessity of making any change, as the parts which are criticised appear again in substantially the same form in Strauch's work. 410. Laurent. A memoir on the Calculus of Variations was written by Laurent in competition for the prize offered by the Academy of Sciences at Paris; see Art. 133. Lam-ent's memoir was sent to the Academy after the time fixed for the reception of the memoirs, but before the judges had published their award. A report on Laurent's memoir is given by Cauchy in the Comptes Rendus ... Vol. 18, pages 920, 921. 1844. We will give a trans- lation of the essential part of this report. The application of the Calculus of Variations to the investi- gation of the maxima and minima values of multiple integrals required especially new formulae of integration by parts and a new notation which should afford an easy expression of these new formula. The judges of the prize had particularly noticed the paragraphs relating to these two objects in the memoir of Sarrus. The corresponding paragraphs in the memoir of Laurent are also worthy of notice. The two authors have employed different methods of establishing the formulae of integration by parts. But the formulas are in reality the same in the two memoirs, although they are expressed by two distinct notations. We may add that when once these formulae are established Laurent uses methods analogous to those of Sarrus in order to obtain the limiting equations. The memoir of Laurent contains besides some observations, which are not without interest, respecting the different ways of verifying the limiting equations. We will not conceal the fact that among the methods employed by Laurent some may be considered rather as methods of induction than as perfectly rigorous methods. But it is generally very easy to verify the exactness of the results obtained by these methods, as the calculations commonly can be easily effected. To sum up we think the memoir of Laurent deserves to be LAURENT. STRAUCH. ROGER. GOODWIN. 477 approved by the Academy, and to be inserted in the Recueil des Savants etr angers. We may add that the memoir does not seem to have been printed as yet. There is a report on two memoirs by Laurent in the Gomptes Rendus ... Vol. 40, pages 632—634. 1855. The report is by Cauchy, and it gives a short account of the scientific labours of Laurent then recently deceased. 411. Strauch. On the sign of the second variation and on relative maxima and minima. Grunert's Archiv der Mathematik und Physik, Vol. 4, pages 39—68. 1844. This article contains some problems in which the second vari- ation of an expression is examined in order to determine whether the expression is really a maximum or a minimum; and some problems of relative maxima and minima values are discussed. All these problems are reproduced by Strauch in his work. 412. Strauch. Remarks on the words tJana^tbn, varzaZ>?e,.... Grunert's Archiv der Mathematik und Physik, Vol. 7, pages 221—224. 1846. This article contains some remarks by Strauch on some of the terms used in the Calculus of Variations ; the remarks are repro- duced by Strauch in his work, Vol. i., pages 69 — 71. 413. Roger. Essay on Brachistochrones. Liouville's Jbwrna? of Mathematics, Vol. 13, pages 41 — 71. 1848. In this essay the author demonstrates several properties relative to brachistochrones. He considers the case when the moving par- ticle is constrained to remain on a surface as well as the case of a free particle. The differential equations of the problem are obtained by the ordinary principles of the Calculus of Variations, and many interesting results are deduced from these equations. 414. Goodwin. Cambridge and Dublin Mathematical Journal, Vol. 3, pages 225—238. 1848. This article is entitled. On certain points in the theory of the Calculus of Variations. The article is chiefly devoted to the expla- 478 GOODWIN. nation of a certain geometrical conception relative to variations. Suppose X and 'y the co-ordinates of a point in a curve. Then it is manifest that we may give the most general infinitesimal vari- ation possible to the position of the point (a;, y) hj giving it a small tangential displacement and also a small normal displace- ment. Let the tangential and normal displacements he denoted "by T and v respectively ; then if t)X and hy be the corresponding displacements parallel to the axes of co-ordinates, and ds an element of the arc of the curve, we have ^ dx ^ dy ts dy ^ dx and these are equivalent to ^ _ dx dy ^ _ ^y ^^ ds ds' ds ds ' The variation of an integral is then expressed so as to involve t and V, and it appears that r does not occur at all in the uninte- grated part, and only once in the integrated part. It is not difiicult to illustrate geometrically the fact that t does not occur in the unintegrated part. The unintegrated part may be denoted by I Uvds, and then the equation U=0 gives the^rm of the curve which is required, and it is manifest that a curve may be made to pass into another which difiers infinitesimally from itself by a normal variation only, and that in fact a tangential variation can have no effect upon the form of the curve, because if a point receive an indefinitely small displacement along the tangent, or which is the same thing along the curve, it still remains in the same curve. The fact that r does not occur in the unintegrated part of the variation of an integral is the principal topic discussed in this article, and it is illustrated and developed in various ways. Three examples are given of the application of the formulae which are investigated. The article concludes with some remarks on the condition of integrability of a function Vdx. In reference to the well-known GOODWIN. YIEILLE. 479 equation which is olDtained as the condition the author says, " I think it would be more proper to saj that the equation ex- presses a condition of Vdx being a perfect differential rather than the condition, for it is nowhere proved that there may not be an indefinite number of other conditions." It must however be re- marked here, that it has been distinctly proved that the equation referred to is sufficient to ensure that Vdx should be integrable as well as necessary/ ; see the last Chapter of the present work. 415. Yieille. Cours complemenfaire d' analyse et de Mecanique ratio7ielle. Paris, 1851. This valuable work contains some investigations relating to our subject. An excellent demonstration of Lagrange's transformation of the equations of motion in Dynamics is given in pages 1 — 9, A chapter entitled Developpements sur le calcul des variations, occupies pages 38 — 50. This chapter contains four articles. (1) The investigation of the maximum or minimum of I Vdx, where V J Xg contains x, y, z and the differential coefficients of y and z with respect to x ; and an equation is given which connects the variables and differential coefficients. (2) To determine the conditions which must subsist among p, q^, r which are all functions of x, y, and z, in order that {pdx + qdy + r dz) may retain a constant value J Wq whatever functions of t may be denoted by x, y, z] the conditions are found to be those which ensure that pdx + qdy -{-rdz \^ an exact differential with respect to x, y, and z, considered as indepen- dent. (3) Having given c?T =^ dx-{- qdy + r dz, where x, y, z are any functions of t, and p, q, r are any functions of x, y, z which may also contain t, it is required to determine under what con- ditions we shall also have ST =p hx-\- qhy -\-rhz', the conditions are found to be the same as in the preceding example. (4) The example just given is now modified by the supposition that x, y, and z are connected by a relation z=F{x, y); the condition now is found to amount to this, that pdx + qdy + rdz must be an exact 480 VIEILLE. CAUGHT. differential after one of the variables has been eliminated by means of the given relation z = Fi^x^ y). A chapter of exercises on the Calculus of Variations occupies pages 113 — 127. Four examples are discussed. (1) Among all curves of given length which are terminated in two fixed points A and B, to find that for which the sum of the products of each element by the square of its distance from the line AB Is a maxi- mum. (2) To find the maximum value of \f{da^-\-d^), subject to the relation that I Aj{dx^ + dy^ + dz^) shall be equal to a given constant. This question admits of easy geometrical treatment, but the process of the Calculus of Variations does not completely succeed, so that Euler's method for solving problems of relative maxima and minima appears to fail. The reason appears to be, as Viellle conjectures, that the second Integral involves a new variable z which is quite independent of the other variables x and y which alone occurred in the first Integral. (3) Assuming that S' along the curve L ; then if the surface 2 be developed the curve L is transformed into a circle. This theorem had been obtained however by previous writers ; see Arts. 427 and 429. (2) Suppose any surface S and on it a curve L of minimum length ; construct a developable sm'face S which touches the sur- face S along the line L ; then if the surface X be developed the curve L is transformed into a straight line. 433. Bjorling. In integrationem cequationis Derivatarum par- tialium superjiciei, cujus in puncto unoquoque principales anibo radii curvedinis cequales sunt signoque contrario. Grunert's Archiv der Mathematik und Physih, Vol. 4, pages 290 — 315. 1844. In the beginning of 1842 Bjorling published a treatise entitled Calcidi Variationum Integralium Duplicium Exercitationes, of which an account has already been given in Art. 311. A large part of that treatise was devoted to the integration of the differential equation which belongs to surfaces of minimum area. In a French scientific journal called V Institute Bjorling saw it stated that Wantzel and Catalan had proved that the only ruled surface of 'minimum area was the heligoide gauche. Bjorling then resolved to reprint his investigations on the integration of the partial differential equation, with some modifications and additions. Thus the present memoir is devoted to the solution of the partial differential equation, and the results obtained coincide essentially with those of the treatise abeady referred to, namely, that of all surfaces of revolution, the only one which satisfies the proposed differential equation is that formed by the revolution of a catenary round its directrix, and that of all surfaces which can be formed by the motion of a straight line which always remains parallel to a fixed plane, the only one which satisfies the proposed dif- ferential equation is the heligoide gauche. Bjorling expresses a hope 'that the demonstrations of Wantzel and Catalan of the state- ment that this is the only surface out of all ruled surfaces which 492 BJOELING. satisfies the proposed diiFerential equation, will soon be published ; Catalan's has since been published, as we have seen in Art. 431. There are two points in the memoir to which we will ad- vert. In a note on x^age 303 Bjorling makes a statement to which he refers more than once afterwards ; it is to the eflfect that if we are seeking the surface for which lldxdT/ ^s/{l +_p^ +q^) is a minimum, and suppose the surface to be bounded by two given curves, the curves must be such that when thej are projected on the plane of {x, y) one projection must be entirely within the other. It is not obvious what he means to be inferred when this condition does not hold, whether he regards the problem as then impossible, or whether he thinks that the ordinary formulse of the Calculus of Variations cannot be applied to it. On page 312 Bjorling considers a certain special example. Sup- pose we have two circles in parallel planes at a distance 2, and sup- pose that the line joining their centres is perpendicular to the planes of the circles, and that the radius of each circle is - ( e + - ] . 2 V e/ Take the line joining the centres as the axis of x and the origin midway betwee*h the centres. Then it might be supposed that the minimum surface would be that formed by revolving round the axis of X, the catenary determined by and taking that portion of it comprised between x= — \ and x—\. But it will be found on trial that the sm-face thus obtained is not necessarily less than that which would be obtained by taking a cylindrical surface round the axis of x as axis with any radius r which is less than 5 (^ +~) > ^^^^ forming the surface of the part of this cylindrical surface which is contained between x = — \ and x=l, together with the plane circular strips at each* end which are necessary to connect the cylindrical surface with the BJOELING. GRUNEET. JACOBI. SCHLAEFFLI. 493 given limiting circles. In fact the area of the surface formed by the revolution of the catenary will he found to be and the area of the surface made up of the cylindrical surface and the plane circular strips is 2^ J2._/+l (, + !)] Now it is quite possible for the former expression to exceed the latter ; for example, the former will exceed the latter by tt [ ^ ) , 3 if r be taken so that r^— 2r + - = 0, that is if r be taken about = "2. o Bjorling brings this forward as an example of the necessity of the restriction he proposed in his note on page 303. It seems to shew no more than this ; a result furnished by the Calculus of Variations must not be assumed to be a maximum or a minimum without investigating the terms of the second order. 434. Grunert. On the Cycloid as the Brachistochrone. Grunert's Archiv der Mathematik und Physih, Vol. 7, pages 308—315. 1846. This article contains an elementary proof of the fact that the cycloid is the brachistochrone, without the use of the Calculus of Variations. 435. Jacobi. On a particular solution of the partial differ- ential equation dW dW d'V_ dx' '^ df^ dz^~ Crelle's Mathematical Journal^ Vol. 36, pages 113 — 134. 1848. In the course of this memoir, Jacobi makes that application of the Calculus of Variations which we have given in Art. 323. 436. Schlaeffli. On the minimum of a certain Integral. Crelle's itfaiAema^zcaZ JoMwaZ, Vol. 43, pages 23 — 36. 1852. 494 SCHLAEFFLI. HOHL. The problem of finding the shortest line on a surface of the second order amounts to making the integral \\/{dx^ + dx^ + dx^) a minimum, where x^^ x^, x^ are connected by an equation of the second degree. In the present memoir the problem considered is to make the integral U{dx^ + dx^+ ... + dx^) a minimum, where the variables x^, x^, ... x^ are connected by an equation of the second degree. The memoir however does not belong to the Calculus of Variations, as there is only one line connected with that subject; in this line the equations for a minimum value furnished by the Calculus of Variations are written down, merely for the purpose of indicating the number of arbitrary constants which should occur in the solution. The solution of the prob- lem considered in the memoir is effected by some complex alge- braical investigations which do not involve the Calculus of Varia- tions. . ' 437. Hohl. Aufgahen zur Lehre vom Grossten und Kleinsten der Bifferenzial-Functionen ... Stuttgart, 1852. This is an octavo volume of 162 pages ; the author is a pro- fessor of Mathematics in the University of Tubingen. The problems are of the same kind as that which we have considered in Art. 3, after Lagrange. Three cases are considered by the author. (1) The maximum or minimum of -F fa?, y, ^j . (2) The maximum or minimum oi F (x,y, -4- , y4 j • (3) The maximum / dz dz \ or minimum ^i Fix, y, z, -j- , -ir) ' Each case is illustrated by the solution of numerous simple examples. The author says that the examples are intended for the exercise of beginners, in Dif- ferentiation, in Integration, and in the higher Geometry. The author says in his preface that he did not become ac- quainted with the work of Strauch b.efore the printing of his own had advanced to the last sheet. He promises if his work is favourably received, to follow it up by a similar collection of HOHL. WITUSKI. JELLETT. 495 examples on the maxima and minima values of integral ex- pressions; the present writer is not aware that this continuation has appeared. ^38. Wituski. De Maximis atgue Minimis valorihus Func- tionum Algebraicarum ... Berlin, 1853. This is an essay written for a degree in the University of Berlin; it contains 25 quarto pages. The essay has no relation to the Calculus of Variations ; it consists of investigations partly respecting the equations furnished by the Differential Calculus for determining the maxima and minima values of expressions, but chiefly respecting the tests for ascertaining whether a maximum or minimum value really exists. 439. Jellett. On the surface which has its mean curvature constant. Liouville's Journal of Mathematics, Vol. 18, pages 163—167. 1853. The Calculus of Variations shews that for a sm'face which includes a maximum volume under a given surface, the mean curvature must be constant. The object of the article is to prove that among all the surfaces whose volume can be expressed by the integral rR rn r2rr r'drsmedddcjy, J J J the sphere is the only sm-face which has its mean curvature constant. The proof depends upon two theorems. (1) For any closed surface see Mr Jellett's Calculus of Variations, page 353. (2) For any closed surface the whole area of the surface 496 BONNET. GRUNEET. SEKEET. the integral being taken over tlie whole surface. This remarkable theorem is proved in the article. 440. Bonnet. Note on the general theory of Surfaces. CoTTvptes Rendus ... Vol. 37, pages 529—532. 1853. This note contains some results relative to the surface of mini- mum area. A new form is proposed for the integral of the differ- ential equation which belongs to such a surface, the new form being in Bonnet's opinion preferable to that given by Monge. Some new properties of such surfaces are enunciated without demonstration. The investigations relative to the integral depend upon a method of considering surfaces which is due to Gauss. Bonnet does not demonstrate the fundamental formulae which he uses. 441. Grunert. On the shortest line between two points on any surface and on the fundamental formulae of spheroidal Trigono- metry. Grunert's Archiv der Mathematik und Physik^ Vol. 22, pages 64—106. 1854. The design of this memoir is to discuss in an elementary manner the subjects mentioned in its title, and there is no reference in it to the Calculus of Variations. 442. Serret. On the least surface comprised between given right lines not situated in the same plane. Comptes Rendus ... Vol. 40, pages 1078—1082. 1855. Legendre asserted that the least surface comprised between two given right lines which are not situated in the same plane is the Mligoide gauche; see Art. 428. Serret shews that this assertion is incorrect, for there is an infinite number of such surfaces, and the Mligoide gauche is only a particular case of them. Serret's investigation is based on Monge' s form of the integral of the differential equation belonging to surfaces of minimum area. 443. Bonnet. On the determination of the arbitrary functions which occm- in the integral of the equation for surfaces of minimum area. Comptes Rendus ... Vol. 40, pages 1107 — 1110. 1855. BONNET. ROGER. 497 Bonnet's design is to shew tliat tlie question discussed by Serret on pages 1078—1082 of this volume of tlie Comptes Rendus ... , and similar questions of greater difficulty, may he investigated by means of the formula which he himself gave in the 37th volume of the Comptes Bendus . . . ; see Art. 440. 444. Roger. Memoir on a certain class of curves. Comptes Bendus.,. Vol. 40, pages 1176, 1177. 1855. This is a brief account by the author of the results of his in- vestigations. It is as follows. We may imagine in space or on a given surface an infinite number of different trajectories which a particle can describe under the action of a given system of forces. Among these trajectories I have considered those which make an integral of the form I ^ {v) ds a minimum, where ^ (v) is a certain function of the velocity, supposed known in terms of the co-ordinates of the moving particle, and s is the arc described from the starting-point. Some curves which have been already studied under various points of view fall under the class which I have defined, and form particular cases of it. The principal are the following. 1. Geodesic lines which correspond to the case for which ^ (v) = a constant. 2. Brachistochrones for which ^{v) =- . 3. The trajectories of least action which are obtained by taking (}){v) =v; these trajec- tories by a well-known principle due to Euler are those which the moving particle is naturally led to describe under the action of the given forces. 4. The lines of greatest slope {h'gnes de plus grande pente) ; these form a peculiar species, which I find corresponds to the case of ,, , , = 0, whatever v may be. The lines belonging to these difierent species and to other species of the same class which have not as yet obtained a definition, or rather a distinctive appellation, possess on the one hand a set of common properties, and on the other hand properties peculiar to the difi'erent species ; the study of these properties ap- 32 498 EOGEE. pears to me to have some interest. The most striking results which I have obtained are the following. I. If we suppose on a given surface a series of trajectories of the same species which start normally from the same curve, and take on each of them arcs described in the same time, the curve formed by the extremities of these arcs will be itself normal to every one of the trajectories, if these trajectories are geodesic lines or brachistochrones, and only in these two cases. (This theorem has already been demonstrated for geodesic lines by Gauss and for brachistochrones by Bertrand.) II. The trajectories of least action, the brachistochrones, and generally the species for which the ratio tA^ vanishes when v=0, are tangents, to the lines of greatest slope, or, which is the same thing, are normals to the curves of level {courhes de niveau), in all the points where the velocity is zero. III. If we suppose the moving particle to be free or to be con- strained to move on a plane, and consider the ratio of the centrifugal force — to the component N of the force estimated along the radius of curvature of the path described by the particle, then 1. For all the curves which make the integ-ral 6 {v) ds a mini- mum the ratio of the component N to the centrifugal force is con- stant throughout the extent of any curve of level. 2. This ratio is absolutely invariable for all the particular species determined by a function of the form ^ [v) = v^, where k is an arbitrary constant, which is in fact the value of the ratio of r 3. In a more special manner this ratio reduces to + 1 for bra- chistochrones and for curves of least action ; so that in these two species the component N is equal, in actual magnitude, to the cen- v^ trifugal force — , and this property belongs exclusively to these ROGER. CATALAN. 499 two curves, including the rigtt line, wiiich may be considered as a variety of either of them. IV. If a curve belongs to two different species it will possess the properties of all the species ; that is, it will be at every point geodesic, curve of least action, brachistochrone, line of greatest slope, &c. For example, in the case of gravity, any meridian of a surface of revolution with its axis vertical. This is the end of the author's account of the results of his investigations. It would appear that these investigations constitute a development of the memoir published in Vol. 13 of Liouville's Journal of Mathematics; see Art. 413. In that memoir Hoger explains what he means by a line of greatest slope and by surfaces of level. It is there stated that the theorem attributed to Gauss was published by him in the memoir in the 6th volume of the Gottingen Transactions. The theorem attributed to Bertrand is there proved by Koger. Roger first supposes the curves to start all from the same point; he says that this theorem was communi- cated to him by Bertrand, and he also gives Bertrand's proof, which is as follows. Suppose a point on a surfece; see figure 12. Let AM, AM', ... be brachistochrones, commencing at the same point A, such that they would be described in equal times by particles starting from A with the same velocity ; then the locus of the points M, M', . . . will be normal to every brachistochrone. For if the angle at M' be acute we can make at M an angle NMM' greater than NM'M; then we shall have MN less than M'N; thus the moving particle having arrived at N with a certain velocity would describe the element NM in less time than it would describe the element NM', its velocity not being sensibly altered while describing the element ; thus the curve ANM would be described in less time than ANM\ that is in less time than AM, which is absurd. 445. Catalan. Note on a surface at every point of which the radii of curvature are equal and of opposite sign. Comptes Rendus ... Vol. 41, pages 35—38. 1855. Catalan proposes to consider whether the well-known differential equation admits of a solution of the form z = j){x) +'v^(?/). He 32—2 500 CATALAN. BONNET. shews that the only solution of such a form is one which in its simplest form may he written z = log cos 3/ — log cos x. He also points out many properties of the surface denoted by this equation. This equation had been noticed before ; see Art. 428. 446. Catalan. Note on two surfaces which have at every point their radii of curvature equal and of opposite sign. Comptes Bendus ... Vol. 41, pages 274—276. 1855. Two surfaces are here given which satisfy the well-known differential equation. One of them is that determined by equa- tion (3) of Art. 428. Catalan points out some properties of this surface. 447. Catalan. On the surfaces which have at every point their radii of curvature equal and of opposite sign. Comptes Bendus ... Vol. 41, pages 1019—1023. 1855. This is an extract from a memoir on the subject named. Some results are given without demonstration. Catalan appears to have transformed the differential equation into forms more convenient for integration than the common form. He is thus enabled to obtain the integral in a more convenient form than Monge's. Some new examples are given of surfaces which have the property considered. 448. Bonnet. Observations on Minima Surfaces. Comptes Bendus ... Vol. 41, pages 1057, 1058. 1855. Bonnet adverts to three notes on the subject of Surfaces of minimum area which Serret had communicated to this volume of the Comptes Bendus . . . Bonnet intimates that his own formulae in the 37th volume of the Comptes Bendus . . . had rendered such investigations superfluous. Bonnet claims for himself the example given by Catalan in his second note, which as we have seen had been given before either of them by Scherk ; see Arts. 446 and 428. Bonnet then adds two more examples of the use of his formulge. On page 1155 of the 41st volume of the Comptes Bendus ... Catalan offers a brief reply to the remarks of Bonnet. This reply was referred by the Academy to the members who had already BONNET. LIOUVILLE. EICHELOT. 501 been appointed to examine Catalan's memoir, namely Liouville, Binet and Chasles. 449. Bonnet. Note on the surfaces for which the sum of the two principal radii of curvature is equal to twice the normal. Com^tes Rendus ... Vol.42, pages 110 — 112. 1856. This is an application of the formulas which Bonnet gave, as he sajs, in the Gomjptes Rendus ... Vol. 37, page 349, to the deter- mination of a class of surfaces which have a remarkable analogy to the surfaces of minimum area. Page 349 seems to be put by mistake for page 529. 450. Bonnet. New remarks on surfaces of minimum area. Compes Rendus ... Vol. 42, pages 532 — 535. 1856. Bonnet says that this article contains a simpler solution than that which he had given in Vol. 40 of the Comptes Rendus ... of the problem to determine the surface of minimum area which touches a given surface along a given curve. 451. Liouville. Remarkable expression of the quantity which by the principle of least action is a minimum in the movement of a system of material particles subject to any connexions. Comptes Rendus ... Vol. 42, pages 1146—1154. 1856. This article is not connected with the Calculus of Variations ; it is interesting in its relation to Dynamics. 452. Richelot. Remarks on the theory of Maxima and Minima. Schumacher's Astronomisclie Nachrichten. No. 1146. 1858. This article relates to the ordinary theory of maxima and minima values of the Differential Calculus. 453. Richelot. On the theory of elliptic functions, and on the differential equations of the Calculus of Variations. Coinptes Rendus... Vol. 49, pages 641—645. 1859. This article states that the differential equations furnished by the Calculus of Variations for the maximum or minimum of an integral may be transformed into other differential equations of the 502 RTCHELOT. LOFFLEE. first order and first degree, which take what the author calls the canonical form ; this term is used because the form agrees with the analogous form in Dynamics. E.ichelot's object is thus the same as that of Ostrogradsky and Clebsch ; see Art. 317. 454. Bode and Fischer. MathematiscTie Lehrstunden von K. H. Schellhach. Aufgahen aus der Lelire vom Grossten und Klein- sten, hearheitet und Tierausgegehen von A. Bode und E. Fischer. I860. This is an octavo volume of 154 pages containing elementary problems not involving the Difi"erential Calculus. 455. We have in Art. 327 referred to some remarks by LofSer as destitute of value; since that article was printed the present writer has seen a later paper by LofSer. This paper is entitled Beitrag zum Prohleme der BrachystocTirone ; it is published in the 41st volume of the SitzungshericJite of the Academy of Sciences of Vienna, pages 53 — 59, 1860. It is remarkable that a scientific society should print a communication with so little to recom- mend it. Loffler's notion is that the limiting equations in problems of maxima and minima are often inadmissible or contradictory, and that in the brachistochrone problem they do not supply sufficient conditions. He takes for example the case in which we require the maximum or minimum value of and supposes that the limiting values of y are not fixed. The term outside the integral sign in the variation of I (y'^ + — ^ ) dx is '^y'^y, and Loffler says that it is equal to ( a^ -1 ) hy, where a^ is a constant. Thus the coefficient of hy is infinite when x = a, and so we cannot make the integrated part vanish at the lower limit. Loffler has not given the coefficient of hy correctly ; for to LOFFLEE. 603 joaake the proposed expression a maximiiin or minimmn, we have the equation a-x ^ ' and this leads to 2y = a^x ■\- a^ — {x — a) log {x — a), where a^ and a^ are arbitrary constants. Thus 2y' =a^ — \ — log {x — a) , and this should he the coefficient of hy instead of what Loffler gives. Nevertheless it is true, as he says, that this coefficient is infinite when a; = a ; this indicates that if the limiting values of y are not fixed the proposed integral cannot be made a maxi- mum or a minimum, and this involves no contradiction and no difficulty. Loffler next considers the brachistochrone problem on the sup- position that the initial point is constrained to lie on one fixed vertical line and the final point on another fixed vertical line. Take the axis of y vertically downwards, and let If we proceed to make U a minimum, we obtain in the usual way V(aJ=V(-4-y)V(l+3/'^ where a^ is a constant. The integrated part of the variation re- duces to and this will not vanish if hy is arbitrary at both the limits unless y vanishes at both limits. Loffler says that this is inadmissible, because the first element of the cycloid must be vertical and not horizontal. There is no reason for saying that the first element of the cycloid must be vertical ; the fact is that our result indi- cates that there is no minimum in the present case ; see Art. 23. There is therefore here no contradiction and no difficulty. 504 LOFFLER. Loffler now takes the general brachistoclirone problem when the initial and final points are constrained to lie on given curves, and the velocity is supposed given at the initial point. He puts down a few of the steps and arrives at the results which we have denoted thus in Art. 300, {pyjr'{x)+ll=0, %'(a?,);?,+ l = 0; therefore '^^'(^2) =%'(^i)' He then asserts, quite untruly, that from the nature of the cycloid, we must have and on this error he constructs a large figure and a corresponding page of text. Lastly, he considers that there are not enough conditions for determining the constants of the problem ; he seems to be in difficulty with respect to the quantity A. But in the case which he has himself considered, A is equal to the value of 1/ at the initial point ; and if A were any given function of the value of 1/ at the initial point the problem could be discussed in a similar manner. Loffler's difficulties arise solely from his own miscon- ceptions. CHAPTER XVII. CONDITIONS OF INTEGEABILITY. 456. The present chapter will be devoted to the subject of the criteria which determine whether proposed expressions are immediately integrable. The history of the subject has not hitherto been fully treated ; and it will be seen that the statements which have been made are deficient in precision. 457. In Gregory's Examples of tJie processes of the Differential and Integral Calculus, first edition, page 285, the relations are given which must hold in order that a function involving two variables and their differentials may be integrable once, twice, thrice,... Gregory says, "these remarkable formulae were first discovered by Euler {Comm. Petrop. Vol. viii.) in his investigations concerning maxima and minima." This does not appear correct; Euler first gave the relation which must hold in order that a function of one variable and its difierential coefficients may be in- tegrable once, but not in the place which Gregory cites. The eighth volume of the Conim. Fetrop. is represented to be for the year 1736, and was published in 1741. It contains a memoir by Euler, entitled Curvarum maximi minimive proprietate gauden- tium inventio; there is nothing in this memoir relating to the conditions of integrability. The eighth volume of the Novi Comm. . . . Petrop. is represented to be for 1760 and 1761, and was published in 1763 ; it has nothing bearing on the conditions of integTability. 458. The tenth volume of the Novi Comm. ... Petrop. is re- presented to be for 1764, and was published in 1766 ; this volume 506 CONDITIONS OF INTEGE ABILITY. contains two memoirs by Euler, connected with the Calculus of Variations. The first memoir is entitled Elementa Calculi Varia- tionum; the second memoir is entitled Analytica explicatio methodi Tnaximorum et minimorum. At the end of the second memoir Euler says : Antequam autem finiam examini Analjstarum egre- gium Theorema subjiciam cujus Veritas ex principiis hactenus positis haud difficulter perspicitur, et quod in Calculo integrali eximium usum prcestare videtur. The theorem is that Zdx is integrable if ijjdU (X/Jb (a/OO (X/JO and that Zdx is not integrable unless this relation holds ; N,P, Q,... being derived from Z in the well-known manner. This appears to be the earliest reference to the Theorem. 459. The third volume of the first edition of Euler' s treatise on the Integral Calculus was published in 1770; the present writer has not seen it, but this date is assigned to it by Strauch in his preface, page x, and the date is confirmed by the testimony given in Vol. 15 of the Novi Comm. ... Fetrop. which will presently be quoted. ^ It appears that the third chapter of the part which treats of the Calculus of Variations contains the theorem, that the necessary and sufficient condition for the integrability of Vdx is that dx dx^ dx^ dx^ " ' the proof given is in substance the same that has usually been adopted in Treatises on the Calculus of Variations. The present writer has not had the opportunity of consulting the first edition of Euler's Integral Calculus, so that he cannot assert positively that the proof is there given. Bertrand, in his Memoir in Cahier 28 of the Journal de VEcole Polyteclmique, quotes Euler's proof but without giving any precise reference. The passage Bertrand quotes occurs in Art. 92, page 425 of the second edition of Euler's Integral Calculus ; the date of the volume is 1793. In the same volume, CONDITIONS OF INTEGEABILITY. 507 Art. 129, page 454, Euler gives tlie form of the variation of I Vdx, where V contains two dependent variables ?/ and z, and their differ- ential coefficients with respect to x. From his result he infers in Art. 131 that two relations must he satisfied in order that Vdx may be integrable, namely, ^_dP d^_d^ d^_ dx dx^ dv? dx^ " ' and a similar relation in connexion with z and its differential coefficients. 460. The fifteenth volume of the Novi Comm. ... Petroj). is represented to be for the year 1770, and was published in 1771. It contains a memoir of 68 pages by Lexell, entitled Be criteriis Inte' grdbilitatis Formida7'um Differentialium. There is a short account of this memoir given in pages 18 — 22 of the volume. In this account Euler's theorem is referred to as, insigne Theorema ab 111. Eulero in Tomo iii. Calculi Integralis allatum, and the following statement is made. Hoc autem Theorema, licet jam demum anno prseterito in nunquam satis laudato opere Calculi Integralis evul- gatum fuit, tamen ad minimum ante 16 annos ab lUustris. ejus Auctore inventum fuisse certissime nobis habemus perspectum. Quum vero interea lUustr. Eulerus hoc Theorema cum insigni quodam Galli^e Mathematico communicasset, probabile omnino est, Illustr. Marchionem de Gondoixet per eum in cognitionem hujus Theorematis pervenisse. Ex Historia enim lUustrissimge Academ. Scient. Parisinffi pro annis 1764 et 1765 accepimus, modo laudatum Marchionem primum demonstrationem hujus Theorematis cum Illustr. Acad. Parisina communicasse, tum vero conscripto Tractatu de Calculo Integrali doctrinam de criteriis integrabilitatis omnino fusius explicasse. It seems singular that in this passage, which claims priority for Euler, it is implied that the theorem was first given by Euler in his IntegTal Calculus in 1770, when we have seen that it was really given by him in the volume of the Novi Comm. . . . published in 1766. Lexell, in his memoir, gives the criteria which determine 508 CONDITIONS OF INTEORABILITY. when an expression admits of integration several times in suc- cession. 461. The volume of the Histoire de VAcademie ...de Paris... for the year 1765 was published in 1768, Here on pages 54 and 55 we find the following statements. M. Le Marquis de Condorcet presented to the Academy a treatise on the Integral Calculus. He solves this problem; given a differential equation of any order with any number of variables, required to determine if this equa- tion in the state in which it is proposed admits or does not admit of an integral of an inferior degree. This important solu- tion is given with all the elegance and all the generality pos- sible. Lacroix, Traite du Gale. Biff. ... Vol. 2, page 238, says "... je passerai aux Equations de condition qu'Euler a rencontrees par line sorte de hasard, et qui ont ^te demontrees pour la premiere fois directement par Condorcet, dont je suivrai d'abord la marche." Accordingly we may presume that Lacroix gives Condorcet's method. The necessity of the condition is shewn very distinctly, and the conditions are given which must hold for a function to admit of integration twice, thrice, &c. 462. The sixteenth volume of the Novi Comm. ... Petrop. is represented to be for the year 1771, and was published in 1772. It contains a memoir by Lexell which occupies 59 pages. Lexell says that he wished to give some examples of the application of the criteria of integrability, and also to give a new demonstration of Euler's theorem, since that which he formerly gave was liable to objection. Lacroix, in his Traite du Gale. Diff. ... Vol. 2, page 249, says, On trouve dans les Novi Gommentarii Acad. Petrop. T. XV. et xvi. deux Memoires dans lesquels M. Lexell s'est propose de prouver la proposition ci-dessus ; mais ses proc^des sont extr^me- ment compliques, et ont paru tels a M. Lagrange. [Legons sur le Calcul des Fonctions, p. 409, de I'editipn in-8° imprimde par M. Courcier, en 1806.) Lagrange's words are quoted in the next article, and they do not bear out the remark of Lacroix ; La- CONDITIONS OF INTEGEABILITY. 509 grange says tliat tlie demonstration in the fifteentb. volume is complicated, and says nothing of the other demonstration, while Lacroix speaks of Lagrange's opinion of loth demonstrations. 463. Lagrange has proved both the necessity and sufficiency of the condition of integrability for the case of a single dependent variable ; and he adds that in the same way the two conditions can be obtained which must hold when there are two dependent variables. -See the Theorie des Fonctions, first edition, page 217; and the Legons sur le Galcul des Fonctions, edition of 1806, page 402. .It is usual on this point to refer to the latter work, but the proof is substantially the same in the two works, though the nature of it is perhaps seen more readily in the former work. On page 409 of the latter work, after Lagrange has given his proof, he remarks, Nous venons de prouver non seulement que la fonction proposde ne peut ^tre une fonction deriv^e esacte, a moins que r equation de condition n'ait lieu, comme Euler et Condorcet I'avaient trouvde, mais encore que si cette Equation a lieu, la fonction sera necessairement une derivee exacte, ce qui restait, ce me semble, a demontrer; car la demonstration qu'on en trouve dans le tome xv. des Novi Commentarn de Petersbourg, est si com- pliquee, qu'il est difficile de juger de sa justesse et de sa generality. 464. In the Legons... Lagrange, after investigating the con- ditions of integrability, gives some examples of their use; see pages 417 — 421 of the work. Suppose in the first place that we have a function of the first order f{x, y,y') ; the condition that it may be an exact differential is /(2/)-[/'(3/')]' = 0. In order that this may be identical /'(y') must not contain y', for if it did [/'(j/)]' would contain y", and as y" would not occur in /'(?/), the whole expression /'(y) — [/' (3/')]' would not vanish identically. Thus f{x, y, y) must be of the form "^ (a^j y)+y' [^, y) ; olO CONDITIONS OF INTEGEABILITY. tlien it will be found that the condition reduces to Next, consider a function of the second order /(a;, 7/,y', y") ; the condition that it may be an exact differential is /(^) -[/'(/)]'+ [/(/)]" = 0. As before, it is necessary that /'(?/") should not contain ?/"; so that /(a?, 3/, y\ y") must be of the form '^{.^,y,y')+y"<^{xyy,y')' Then it will be found that the equation of condition will become t' iy) + y" ^' {y) - W (3/')]' + W («^)]' + b'<^' (y)]' = 0. Let ^'{x) +y4>'{y) —'^'{y') be denoted by % [x, y, y), so that the condition becomes -f' (3/) +y"<^' (y) + [x {^^ y^ 2/')]' = ; that is t' iy) + x{^) + y'x (y) + y" l^' (y) + x iy')] = 0. And y" does not occur in any of the functions -^'(y), xi^)^ xiy)} so that the last equation cannot be identically true unless '{y) + x'iy') = o, and ^'{y)+xi^)+y'x'iy)=^' - Lagrange adds on page 421 — In like manner as in the case of a function of the second order, the equation of condition decomposes into two which must hold simultaneously, so it may be proved that for a function of the third order it will decompose into three, for a function of the fourth order it will decompose into four ; and so on. This statement has been developed in two elaborate memoirs by Raabe and Joachimsthal. Raabe's memoir is in Crelle's Mathematical Journal, Yo\. SI, pages 181 — 212. 1846. Joachimsthal's memoir is mCreHQ's Mathematical Journal, Vol.33, pages 95— 116. 1846. CONDITIONS OF IN TEGE ABILITY. 511 465. As we have already stated, Lacroix gives a proof of the necessity of the conditions of integrability. His method is inde- pendent of the Calculus of Variations. But he does not prove the sufficiency of the conditions by this method, but refers to the Cal- culus of Variations on this point. Accordingly he returns to the subject in the chapter on the Calculus of Variations, and there he improves, as he considers, Euler's proof; see the Traite du Calc. Dig. ... Vol. 2, pages 249 and 764. 466. The fourteenth volume of Gergonne's Annates de MathS- matiques contains a memoir on the integrability of differential expres^sions by M. F. Sarrus ; the date of publication is January, 1824. The memoir occupies pages 197 — 205 of the volume. Sarrus begins by referring to the remarks of Lagrange which we have quoted in Art. 463. He then proves that the conditions of integrability are necessary; he takes the case in which two variables x and y are functions of a third variable t, and an ex- pression involves x and y and their differential coefficients. In proving that the conditions are necessary, Sarrus adopts precisely the same method as Lacroix, but he does not give any reference to him or to Condorcet. Sarrus then proves that the conditions are sufficient. The demonstration given by Sarrus is perhaps the best for elementary purposes that has yet appeared, unless it be considered preferable to prove the necessity of the conditions in the manner given by Sarrus, and the sufficiency of the conditions in the manner given in Moigno's Legons de Calc. Diff. et de Gale. Int. 467. Another memoir on the conditions of integrability ap- peared in the fourteenth volume of Gergonne's Annales ... , pages 319—323. The question considered is the following. Suppose V a function of X and y and their differential coefficients with respect to a third variable t. Then the two conditions which must hold in order that Vdt may be integrable are known from the memoir of Sarrus. Now suppose that y is made a function of x, it is obvious that a 512 CONDITIONS OF INTEGEABILITY. single condition would ensure the integrabilitj of Vdt ; it is required to find tliat condition. The result is dt^ dt ' where X and Y are the functions which we should have to equate to zero to ensure the integrahility of Ydt if y had not been made a function of a;. This result is obtained by simple transformations. The result mav be easily obtained by the Calculus of Variations ; for if y be not supposed a function of x, we obtain in the ordinary way for the unintegrated part of S / Vdt the expression suppose y is made a function of a?, then this term becomes Thus in order that Vdt may be integrable, we must have dy ax dt At the end of the memoir the writer says that the condition is exactly that of Lagrange, Legons ... page 412 of the edition of 1806. But Lagrange has there a different question before him; Lagrange's result is in fact that which we have noticed in Art. 93, and have expressed thus, Ey+Kz' = 0. 468. Graeffe briefly refers to the condition of integrability on page 46 of his essay ; see Art. 306. He quotes the theorem of CONDITIONS OF INTEGEABILITT. 513 Euler as we have given it in Art. 458, and says, Manifesto Eulerus ad illam gequationem in qugestionibus quae ad calculum variationum spectant, instituendis venit, unde accidit, ut his principiis theorema superstrueret. Sed ejusdem evidentia adhuc desiderabatur et quan- quam Condorcet et Lexell demons trationem in solius calculi inte- gralis notionibus fundatam tentabant, Lagrange tamen primus rite confirmavit, si formula V evanescat, semper quantitatem Zdx inte- grari posse. Graeffe refers to Condorcet, du Calcul Integral, p. 16 seq^. Novi. Comm. ... Petrop. T. XV. p. 127. Lagrange Legons ... p. 401 seq. 469. In Poisson's Memoir on the Calculus of Variations, pages 260 — 270 are devoted to our present subject; see Art. 96. Poisson first shews very briefly the necessity of the condition. He says that if Vdx is an exact differential the integral ZJwill be a function of iCg, 2/q, y^, yl\ ... yl, y", ... ; thus the value oihXJ must reduce to the part F, and therefore the factor H under the integral sign must vanish ; see equation (3) of Art. 86, Poisson adds the follow- ing words : " Thus the same equation S= which determines the value oiy corresponding to the maximum or minimum of U, when Vdx is not an exact differential, must become identical when Vdx is an exact differential. This remark is due to Euler, who has thus been the first to express by an equation the necessary condition for the integrability of a diff'erential formula of any order. Lagrange has proved by means of very complicated series not only that the equation H= is necessary, but that it is sufficient for the integrability of Vdxi, Legons ... page 409 of the edition of 1806." Poisson then says that he will give a demonstration of the second part of the proposition which appears more simple to him, and which has the advantage of presenting the integral of Vdx under a finite form, when the condition H= holds. 470. A note by Sarrus is given in the Comptes Rendus.. . Vol. I. pages 115 — 117, 1835. This note enunciates some results, which the author had obtained as generalisations of his memoir in Ger- gonne's Annales.... 471. A memoir by Dirksen on the conditions of integrability of functions of several variables occurs in the volume for 1836 of 33 514 CONDITIONS OF INTEGEABILITY. the Transactions of the Academy of Sciences of Berlin ; the date of the vokime is 1838. This memoir names Euler, Condorcet, Lexell, Lagrange and Poisson. Dirksen agrees with Lagrange in speak- ing unfavom-aUy of Lexell's first memoir ; and Dirksen adds that Lexell's second memoir, which Lagrange does not mention, is un- satisfactory. Dirksen objects to Poisson's proof, because it depends on the Calculus of Variations, and intimates that a proof depending upon considerations which are not foreign to the subject, is still required. Accordingly, he supplies some tedious investigations on the subject ; he proves both the necessity and sufficiency of the con- dition, considering the case of one variable. 472. A memoir by Bertrand on the conditions of integrability of differential functions was published in the Journal de VEcole Polytechnique, Cahier 28, 1841, pages 249 — 275. Bertrand infers from the words of Lagrange and Poisson that they did not know that Euler had professed to prove the sufficiency as well as the necessity of the condition. Bertrand quotes Euler' s words as we have already stated in Art. 459. After some remarks on the history of the subject, Bertrand's memoir is divided into three sections. In his first section, Bertrand proves the necessity and sufficiency of the condition. He says himself that his proof agrees with Euler's when the latter is so modified as to be placed beyond the reach of objection. He then shews how to effect the integi-ation when the condition is satisfied. Bertrand then investigates the conditions when a function is to admit of successive integration ; next he considers the case when there are two dependent variables ; and lastly, he considers the condition which must hold in order that 1 1 Vdxdy may be capable of expression without assigning any par- ticular relation between z, x and y, where F is a function of x, y, z, and the differential coefficients of z with respect to x and y. All these investigations are simple and conclusive. Bertrand begins his second section by saying that his demonstra- tion in the first section depended entirely on the Calculus of Varia- tions, and so he says, diff^re en cela de celles qui avaient dte pro- poshes jusqu'ici par Lexell, Lagrange, Poisson, et derni^rement CONDITIONS OF INTEGRABILITT. 515 encore par M. Sarrus. These words would suggest to a reader that the memoir of Sarrus was subsequent to that of Poisson, which we know, however, is not the case. Bertrand adds that these mathe- maticians establish the sufficiency of the condition by effecting the integration, the possibility of which they wish to prove, and he says that they seem to regard this as the only difficulty in the question. He considers all the demonstrations which have been given very complicated, and thinks he has found a simple demonstration. Ac- cordingly, he establishes the sufficiency of the condition. His proof is, as he says, founded on the same principle as Poisson's, but it avoids the use of the Calculus of Variations. Bertrand's proof is a simplification of Poisson's. Bertrand next proves the necessity of the condition ; this proof seems rather difficult but decisive. In his third section Bertrand gives some interesting applications to Mechanics. 473. The second volume of Moigno's Legons de Gale. Diff. et de Gale. Integ. is dated 1844. Moig-no refers to our present subject on page xxxvii. of his preface, and considers it on pages 550 — 563 of the work. Moigno states that Lexell, Lagrange, and Poisson seem not to have been aware that Euler had proved not only that the condition is necessary, but that it is sufficient. This seems in- correct so far as Lexell is concerned ; for Lexell says that his object was to give a proof without using the Galeulus of Variations, so that he appears to imply that the proposition had been established by the: use of that calculus. • Moigno's proof was communicated to him by M. Jacques Binet. The method of proving the sufficiency of the condition may be de- scribed as an improvement on Bertrand's simplification of Poisson's proof. The proofs of Poisson and Bertrand are liable to failure, be- cause a certain quantity which occurs may become infinite or inde- terminate ; the proof given by Moigno is free from this difficulty. The proof of the necessity of the condition given by Moigno seems open to an objection urged by Professor De Morgan in a memoir which we shall presently notice. Mr De Morgan says : — " Again, it is to be shewn, not only that th& criterion is sufficient, but that it is neeessary. Some of the proofs of the latter point 33—2 516 CONDITIONS OF INTEGEABILITY. appear to me to fail entirely. They depend upon the reduction of Vdx to an integrated portion together with an integral of the form / l{Vy— VJ + ...) Q dx. This, it is assumed, must vanish ; which though clear enough in the common case in which Q=y, and Vy—... is a function of x only, is not sufficiently supported in any other. Why may not {Vy— ...) ^ be a new integrable function?" It does not seem that this objection holds against any other proof besides that given by Moigno. Both Bertrand's proof and that given by Moigno of the suffi- ciency of the conditions allow us to draw the two inferences drawn by Poisson ; see Art. 96. 474. An article on the integrability of functions by Professor Bruun, of Odessa, was published in 1848, in the eighth number of the seventh volume of the Bulletin... Physico-Matliematique of the Academy of St Petersburg ; the article is in German. This article proves both the necessity and sufficiency of the condition ; the proof depends on the Calculus of Variations. The method resembles Poisson's, but is much simpler. This article is included in Pro- fessor Bruun's Manual of the Calculus of Variations. 475. We may now refer to some investigations by Bertrand and Sarrus which are connected with the present subject. Ber- trand's investigations were mentioned in the Comptes Rendus ... Vol. 28, pages 350, 351. 1849. Sarrus gave on pages 439 — 442 of the same volume a brief account of the method which he had for many years explained in his lectures, and which he presumed would be found to agree with Bertrand's. A memoir by Bertrand explaining his method was published in Liouville's Journal of Mathematics, Vol. 14, pages 123 — 131. 1849. This memoir is followed by a note by Sarrus, which occupies pages 131 — 134 of the volume. The method of Bertrand and Sarrus is different from that of previous writers on the subject. Bertrand's own words will give an idea of it. After referring to Euler's well-known condition of integrability, which had been so often demonstrated, Berti-and CONDITIONS OF INTEGEABILITY. 517 makes the following remarks. Notwithstanding the elegant form of this condition the application of it is very laborious. In order to make use of the condition, we have to perform a large number of diflferentiations, and when the condition is satisfied we have to perform a new set of operations in order to obtain the integral which is thus known to exist. The method which I propose in this memoir dififers widely from that of Euler, and it would require some complicated investigations to establish their agreement in a direct manner ; the method does not certainly lead to such an elegant condition as Euler's, but the operations which it requires have the great advantage of simplicity. It is by integrating a proposed function that we ascertain that it is integrable; each operation is followed by a verification, and we are relieved from the necessity of continuing the process if the verification does not succeed. We have thus an advantage analogous to that of the method of commensm-able roots in the Theory of Equations; for this method, although it does not give us a formula for the roots, indicates a series of operations by which we may find these roots, and a single operation will often shew that such a root does not exist. We may add that the method is explained in Professor Boole's Differential Equations, pages 219 — 222. 476. Minich. An article on the present subject occm's in Tortolini's Annali di Scienze Matematiche e Fisiche, Vol. 1, pages 321 — 336. 1850. The article is said to be an extract from an unpublished memoir. The article is divided into three sections. In the first section Minich proposes to exhibit the conditions which ensure that a function shall be susceptible of repeated inte- gration, under a simpler form than the well-known form. An example will give a clear idea of Minich's object. Suppose we have an expression V which involves x and y and the differential coefficients of y with respect to x up to -^ ; and let the partial differential coefficients of V with respect to y, ~- , -j^ > y^ and -t4 ; be denoted by N, P, Q, B, 8 respectively. Then the 518 CONDITIONS OF INTEGEABILITY. conditions 'whicli are necessary and sufficient in order tliat F may be immediately integrable four times in succession are known to be dx dot? dx^ dx^ ' ax ax ax ^ ^dB ,d'S ^ 5-4^ = 0. . ax Minich substitutes for this system the following more simple system, iN-^=0, 3P-2^ = 0, 2C-3^ = 0, ^-4^ = 0. ax ax ax ax If we only require that V shall be immediately integrable three times in succession, the conditions will consist of the first three of the first system given above ; and Minich substitutes for them the following, dP d^Q 6iV - 3 ;:^ + -7-x = 0, 3P-4^ + 3^ = 0, ax ax dR . d'8 And similarly if V is to be immediately integrable twice in succession, Minich gives the two conditions, dx dx^ dx? ' dx dx^ dx^ CONDITIONS OF INTEGEABILITY. 519 ' Thus in eveiy case the last condition of Minich's system is the same as the last condition of the ordinary system, and tlie other conditions of Minich's system are simpler than the conditions of the ordinary , system. Minich gives a general investigation, and shews that the ordinary system can be deduced from his system. He does not shew conversely that his system can he deduced from the ordinary system ; this however is the case, and it can be easily verified in the example which we have given. The object of the second section of Minich's article may be seen from an example which occurs in it. Suppose we require the con- dition which must hold in order that a given expression Rdx^ + Sdx dy + Tdy^ may result by differentiating an expression of the form Pdx-\- Qdy, on the supposition that dx and dy are both constant. The required condition is found to be d'R d'S d'T_ dy^ dxdy dx^ The third section of Minich's article relates to the integration of expressions in Finite Differences. Lacroix intimates that Con- dorcet was the first to consider this subject, and Lacroix considers the subject more curious than useful; see the Traite du Cole. Diff. et du Gale. Int. Yol. 3, page 311. Minich investigates the condition which is necessary in order that one immediate Finite Integration may be possible. Suppose V any function of x, y, Ay, A^y, ... A"y ; let Ay be denoted hj jy^, and A^y by^^' ^^^ so on. Let the symbol B be equivalent to 1 + A. Then the necessary condition is E-^-AE'^-'^^- A'E''-^ — - + (-l)»A'^4^=0. dy dp^ dp^ dp^ This condition may also be put in another form. Suppose that in V we put for Ay, A^y, . . . their values in terms ^^y^yi^y2^ -•; namely ^y=y,-y^ a'3/ = ^2-%i + ^' ••■• 520 CONDITIONS OF INTEGEABILITT. then F becomes a fanction o^ x,y,y^,y^i...yn' The condition may now be expressed thus, dy dy^ dy^ dy^ dV . In this form of the condition -7- is not the same thing as was denoted by -j- in the first form of the condition. Minich then briefly indicates the conditions necessary in order that it may be possible to effect immediate Finite Integration any number of times in succession ; and he shews that the system of conditions which he first obtains is deducible from a second system which is more simple, so that this part of the third section is analogous to the first section. 477. In Mr Jellett's treatise on the Calculus of Variations a chapter is devoted to the present subject. The ordinary proof by the Calculus of Variations of the necessity and sufficiency of the condition of integrability is given, and then five propositions are discussed. (1) To investigate the conditions under which a func- tion will admit of immediate integration m times successively. (2) To find the form of the function V in order that 1 1 Vdx dy may be reduced to a single integral, when F is a function of a?, y, z, p^ and ^. (3) To find the form of the function F in order that 1 1 Vdx dy may be reducible to a single integral, when F is a func- tion of X, y, z, p, q, r, s, and t. (4) To find whether it is possible to represent the superficial area of a surface by any such formula as T+ !Jf{p, e, (j))ddd(jy, where F is a quantity referring solely to the limits of integration, P is the perpendicular from the origin u.pon the tangent plane, and 6 and <^ are the polar angles which determine the position of this perpendicular. (5) Let E and B' be the principal radii of curvature CONDITIONS OP INTEGRABILITY. 521 of a closed surface, P the perpendicular on the tangent plane, and day the element of the spherical surface described by a portion of this perpendicular whose length is equal to unity. Then jj{R + B')dco = 2JjPda>, the integrals being extended throughout the entire of the closed surface. 478. A memoir On some points of the Integral Calculus by Professor De Morgan was published in 1851 in the second part of the ninth volume of the Transactions of the Camhridge PMlosopJiical Society. The fourth section of the Memoir is devoted to the con- dition of integrability of a differential expression. After the memoir had been read before the Society Mr De Morgan became acquainted with the memoir of Sarrus, which we have noticed in Art. 466 ; but as Mr De Morgan's copy of this memoir was detached from the volume to which it belonged, he did not know in what journal it had been published, and made a wrong conjecture. Mr De Morgan says with respect to Sarrus's memoir, " This memoir contains the proof here given, in substance, though the equations on which the condition is founded are not demonstrated. It is singular that M. Bertrand takes no notice of it, except to observe that M. Sarrus does not use the calculus of variations. MM. Cauchy and Moigno pass it over altogether. But it must be observed that M. Sarrus establishes only the necessity of the condition, and does not esta- blish its sufficiency, except when the equations that give it are presented with it." The statement that Sarrus does not prove the sufficiency of the condition is incorrect. By " MM. Cauchy and Moigno" is meant the work published under the name of Moigno which we have noticed in Art. 473. It is not obvious what is meant by the remark that "the equations on which the condition is founded are not demonstrated." 479. There is a very good elementary discussion of the subject in Stegmann's treatise on the Calculus of Variations, pages 118 — 132. Stegmann begins by remarking that the equation fur- nished by the Calculus of Variations for the maximum or minimum 522 CONDITIONS OF INTEGEABILITY. of an integral may in some cases be impossible and in some ' cases identical. An instance of tlie first kind is supplied by endeavour- ing to find tlie maximum or minimum of 1 [xp — y) dx. Here we' should obtain as the condition for a maximum or a minimum d — l—-j-x-=0, that is —2 = 0, which is impossible. In fact if we transform the proposed expression to polar co-ordinates we find that we are requiring the maximum or minimum of jr^dO, and it is obvious that this function may be made either as great as we please or as small as we please. Stegmann then passes on to. the case in which the equation becomes an identity, and this leads him to discuss the condition of integrability. He proves the necessity of the condition- in the same way as Sarrus, and the sufficiency of the condition in the same way as Binet in Moigno's work. Stegmann makes a remark on his page 123 which we will give here. Suppose Vdx a perfect differential of u, where u involves x and 1/ and the differential coefficients of y with respect to x up to J . Let y, stand for ^ . Then , „ dV d^u d^u d\ d^u therefore --=^^ + ^y, + ^^^2,,+ ...+^^^ 3,^, Tin ^^ _ d du dy dx dy'' where the right-hand member means the complete differential co- efficient of -J- with respect to x. Therefore -^ = I -y- dx, dy J dy ' that is -J- I Vdx — I -7— dx, -CONDITIONS OF INTEGRABILITY. 523 SO that if Vdx is a perfect differential, the two operations of com- -plete integration with reapect to x and partial differentiation with respect to ^, may he performed on V in either order. 480. We will close this chapter by giving a translation of the memoir of Sarrus which we have noticed in Art. 466, and also an account of the method adopted by Bruun which we have noticed in Art. 474. 481. The present article is a translation of the memoir of Sarrus. The investigation of the conditions of integrahility of differential functions which has chiefly engaged Euler and Condorcet constitutes one of the most important branches of the higher analysis. The method of variations leads very simply to these conditions, but the use of this method in investigations which strictly belong to the Integral Calculus seems indirect, and moreover it does not assist us in arriving at the integral when these conditions are fulfilled. Euler and Condorcet proved satisfactorily by their analysis that the conditions which they obtained are necessary ; but Lexell appears to be the first who without using any considerations foreign to the integral calculus, tried to demonstrate that these conditions are sufficient, that is, that they assure us of the possibility of effect- ing the integration, which is the important point in the theory [Novi Comm. ... Pet. Vol. xv). Unfortunately, as Lagrange re- marks, the demonstration of Lexell is so complicated that it is difficult to judge of its accuracy and its generality. In reflecting on this subject it appears to us that the processes of the differential calculus, strictly so called, are sufficient by them- selves to lead in a simple manner to the conditions of integrability and to the demonstration of the important proposition of Lexell ; and this we propose to shew in this brief memoir. In all that follows x and y will be any functions of a third variable, the differential of which we shall take for unity, and of any number of constants. For abridgement, we shall represent dx,d^x,d^x,... hjx^,x^,x^,...,^n^dy,d^y,d^y,...}ijy^,y^,y^,...', 524 CONDITIONS OF INTEGRABILITY. P, Pj, Pg, P3, ... will be any functions of a?, x^, x^, x^, ... '^"t-i' 2/, yi» 2/2' ^3' — 2^n-i» and their differentials will be re- spectively ^, 2>i, p^, Ps,'-' Thus we have identically, _ ,._ ^=d— ^-d— — it-d— ^— ^ . ^ aa^j-^^ ^^i+2 '^■^i+3 '^'^Mi wliicli sliews that u^^^ is entirely of the same nature as w^. Let us now suppose that u is a function of £c, a^^ , a?^ > • • • > ^m > ?/, 3/1 5 2^2 ' '•• iVn') which satisfies the condition aa? aajj aajg ax^^ hj operations analogous to those which gave us equation (8) we shall obtain u = p + Uj^, «*! = A^i+A + ^25 u^ = A^x^+2ys + u„ w„, = ^,„a;„, +^,„ + Y, Y being a function of t/, 7/^, z/^, ... , y^ only. Add these equations and put for abridgement q = A^x^ + A^x^ + A^x^+...+A^x^ + Pi+ A+--- +i>m; thus w = 2' + -^j in which q is evidently an exact differential because each of the terms of which it is composed is an exact differential. . If u did not involve y and its differential coefficients t/^^ , y^ , ^3: •••? y«> the function which we have represented by I^ would be a constant and therefore zero, otherwise u would be composed of heterogeneous terms, which can never be the case ; thus u would then be an exact differential. 528 CONDITIONS OF INTEGEABILITY. If on the contrary u involves y and its differential coefficients ^1 5 ^2 5 2^3 J '•' i yn) l>^t so tHat the following equation is identically- satisfied, = T d-j — \-d -J ... ± a -J— > dy dy^ dy^ dy^ the function Y may be different from zero ; but by substituting in this equation the value of u just given, and observing that since c[ is an exact differential we have 0^%-d^ + d^^- +c?»^, dy dy^ dy^ dy^ we obtain, by reduction, dy dy^ dy^ - e%„ ' from this we conclude as before, that since Y only involves y and its differential coefficients y^, y^, y^, ••' ■> yn: this function Fis neces- sarily an exact differential, so that in. this case, as in the preceding, u is still an exact differential. In order to simplify the question we have supposed that all the functions involved only two variables x and y and their differential coefficients ; but it is easy to see that the question would not be- come very much complicated if we wished to consider more than two variables, and that moreover the conclusions would be abso- lutely the same. [It would perhaps have been clearer if Sarrus had explicitly introduced the third variable, say t, of which x and y may be sup- posed functions ; thus in his value of p we should add a term on dP the right — ^ ; his equations (1) and (2) would still hold. His method really amounts to the following ; let V be any function of a, y, t, and the differential coefficients of x and y with respect to t ; then suppose Vdt separated into two parts, first, that part which would arise from supposing t variable, but not x, y, and their differ- ential coefficients, secondly, that part which would arise from regard- CONDITIONS OF INTEGEABILITY. 529 ing X, y and their differential coefficients as variables. Then the first part may be supposed obtained by ordinary explicit integration, and Sarrus disregards it. Dirksen's process, which we have referred to in Art. 471, resembles that of Sarrus in this respect; both in fact follow Condorcet's method as given by Lacroix.] 482. We will now give an account of the method adopted by Bruun which we have noticed in Art. 474. Bruun proves the necessity of the condition in the same way as it is usually proved in works on the Calculus of Variations. His proof of the sufficiency of the condition is substantially the fol- lowing. Let F be a function of x and y and the differential coefficients of y with respect to x, which satisfies the condition of integrability, say V=f{x, y, y', y", ...). Change y into y + tSy, and let Vt denote what V now becomes, so that Vt =f{x, y + tSy, y' + %', y" + fBy", ...). Then let ^~ j ^^^^j so that dU fdV, dt J dt Now -7-' will consist of a series of terms which we may de- note by Lhy + Mhy' + Nhj" + P^y" + . . . Apply the process of integration by parts in the usual manner of the Calculus of Variations, and we shall obtain dt " \ dx dx^ + V(^- ) + [^ f^ dM d'N d^P \ , 34 530 CONDITIONS OF INTEGEABILITY. The part under the integral sign vanishes, because the con- dition of integrabilitj is supposed to be satisfied with respect to f{x,y,y',y'\ ...), and it will therefore be satisfied when y is changed into y + thy. Thus we may express our result as follows, ^ = % t (a., 2, + %, y' + %', y" + tZy" , ...) + V'^i {^^ y + %» y' + %'' y" + %"» • • •) + ¥>2(^.2/ + %. y + %', 2/" + %",.-.) + Integrate with respect to t from < = to < = 1 ; then the left- hand member gives us U^— U^, so that \f[x, y + hy, y + S/, ?/" + hy", ...)dx -jf{x, y, y', y", ...)dx = 1 hyy^{x,y + tBy, y' + %', y" + thy", ...) + %' -^^i {^, y + %. 3/' H- %'' 2/" + %"» •••) + S3/"'»^2 {^^ y + %5 3/' + %'j 3/" + %"» • • •) + \dt. In this result put for y and y for 8?/ ; thus J/(a^> 2^: y'j y"^ •••) dx- jf{x, 0, 0, 0, ...) c?a; = 1 |S3/'.|r (aj, «3/, ^y, ty", ...)+Sy'^|r^ {x, ty, ttj, ty" , ...) + %" ^2 (^, ^2/, ^2/'. iy'\ • • + U^. This is in fact the result originally obtained by Poisson ; see Art. 96. INDEX OF NAMES. The mimhers refer to the pages of the Volume. Abbatt, 460. Aiiy, 442. Ampere, 440, 470. Arndt, 487. Bertrand, 37,258—265,311—315, 346, 348, 514, 516. Bjorling, 343—345, 491—493. Bode, 502. Bonnet, 315—320, 349, 496, 500, 501. Boole, 474. Bordoni, 442—448. Brasclimann, 482. Briosclii, 282—284. Brunacci, 233—242, 436—438. Bruun, 461—463, 516, 529. Bnqnoy, 335. Busse, 484. Carmichael. 481. Catalan, 490, 500, 501. Caucliy, 210—228, 480, 481. Choisy, 334. Clebscli, 308—310. ' Cournot, 461. Crelle, 10, 471, 473, 474. Delaunay, 140—181, 254—258, 348, 475. De Morgan, 460, 521. Dirksen, 28—30, 513. Eisenlohr, 284—293. Euler, 11, 16, 506. Fischer, 502. Gauss, 37—52. Gergonne, 439, 471—473. Giesel, 367. Goldschmidt, 340. Goodwin, 477—479. Graeffe, 335, 512. Grunert, 493, 496. Hall, 461. Heine, 321—325, 366. Holil, 494. Hornstein, 350. Jacobi, 243—253. Jellett, 401—412, 495, 520. Knight, 485. 532 INDEX OF NAMES. Lacroix, 11 — 27, 439. Lagrange, 1 — 10, 509. Laui-ent, 476. Lebesgue, 254. Legendre, 229—233, 489. Lehmus, 486. Lexell, 507. Lindeloef, 368—372. Liouville, 501. Loffler, 367, 502—504. Mainardi, 265—281. Meyer, 466—469. Michaelis, 489. Minding, 325, 336—339, 486, 490. 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A set of Macmillan §• Co.'s Class Books will be found in the Educational Department {Class 29) of the International Exhibition, and for which a Medal has been awarded. Arithmetic. For the use of Schools. By Baenaed Smith, M.A., Fellow of St. Peter's College, Cambridge. New Edition. Crown 8vo. cloth, 4s. 6d. A Key to the Arithmetic for Schools. By Baenaed Smith, M.A., Fellow of St. Peter's College, Cambridge. Second Edition, Crown 8vo. cloth, 8s. 6d. Arithmetic and Algebra, in their Principles and Ap- phcation : with numerous systematically arranged Examples, taken from the Cambridge Examination Papers. By Baenaed Smith, M.A. Fellow of St. Peter's College, Cambridge. Eighth Edition. Crown 8vo. cloth, 10s. 6d. Exercises in Arithmetic. By Baenaed Smith. With Answers. Crown 8vo. limp cloth, 2s. 6d. Or sold separately, as follows : — Part I. Is. Part II. Is. Answers 6d. An Elementary Treatise on the Theory of Equations, with a Collection of Examples. By I. 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Crown Svo. 6s. 6d. 15 Senate-House Mathematical Problems. With Solutions 1848-51. By Feeeees and Jackson. 8vo. 15s. 6c1. 1848-51. (Riders.) By Jameson. Svo. 7s. 6d. 1854. By Waiton and Mackenzie. Svo. 10s. 6d. 1857. By Campion and AVaiton. Svo. Ss. 6d. 1860. By Route and Watson. Crown Svo. 7s. 6d. Hellenica: a First Grreek Reading-Book. Being History of Greece, taken from Diodorus and Tliucydides. By Jos Weight, M.A. Second Edition. Pp. 150 (1857). Fcp. Svo. 3s. 6a. Demosthenes on the Crown. With English Notes. By B. Deake, M.A. Second Edition, to which is prefixed iEschines aarainst Ctesiplion. With Enghsh Notes. (1860.) Pep. Svo. 5s. Juvenal. For Schools. With English Notes and an Index. By John E. Matoe, M.A. Pp. 464 (1853). Crown Svo. 10s. 6d. Cicero's Second Philippic. With English Notes. By John E. B. Matoe. Pp. 16S (1861). 5s. Help to Latin Grammar ; or, the Form and Use of Words in Latin. Witli Progressive Exercises. By Josiah AVeight, M.A. Pp. 175 (1855). Crown Svo. 4s. 6d. 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