88 VARIATIONS Again, the equation at the limits is 1 s i fj.ds+ Q ............ (30) ; also at each limit we have dx = 8x + xds, dy = Sy + yds ; hence (30) / i / *i transforms into / (/* - }ds + / (xdx + f/dy) = 0,
or ed(s 1 -s <) } + / (xdce + j/dy} = (31). S Now if the length of the curve be given, the first term in (31) vanishes ; whereas, if the length be not given, we must have c = 0. Consequently in the isoperimetrical problem we have (ji = + c, and in the general case = /u. If we substitute /a for A in (28) we can readily deduce the results already given in 13. 23. More generally, let /a be a function of x, y, x, and if, then we have V n + ^(x 2 + y 2 - 1). Here we readily deduce .dV .dV r=c + x- r - + y-rr (32); ax ay and the equation at the limits becomes This shows, as before, that when the length of the curve is not given we have c 0. 24. The criterion for distinguishing between maximum and minimum solutions was investigated by Legendre, Lagrange, and other eminent mathematicians, for whose contributions to the solution of this problem, see Tod- hunter s History of the Calculus of Variations. The full analytical investigation was, however, first given by Jacobi, of whose results we here give a brief discussion. In this investigation, as in ordinary problems of maxima and minima, it is necessary to take into consideration the squares and the higher powers of our indefinitely small variations. Thus, if U= I Vdx, the complete variation of ?7may be written 8U+fU+ &c., where 5 U depends on the first powers of 8y, 5y, &c., and 6 2 7 on their second powers, &c. Again, we will suppose that the limiting values of y and of its first n - 1 differential coefficients are fixed, then 1 - r x i p 8Vdx= M8ydx,a,ndd U= I SMSydx. The solutions of --0 *^0 *^0 the problem, as already observed ( 8), are given by the equation J/=0. Also for a true maximum it is necessary that 8~ U should be constantly negative, and for a minimum that it should be positive. If 5 2 7=0, or if it change its sign between the limits of integration, the result is in general neither a maximum nor a minimum. As before, we suppose V a function of x, y, y . . . y( n and dx therefore dp
dy dy
dy dy (33). J When this is developed, we see that every term, disregarding sign, d V is of the form D - - r D m Sy. Also, if we combine this with dy^ dyV ll > the corresponding term, and make l=m + r, we readily see that ur. - Hence 8M is reducible to the form 8M=(A + DA 1 D + D*A.,D 2 + . . . + D n A n D n }8y ...(35), provided it can be shown that D r u + ( -l) r uD r is reducible to an operation of such a form. Now this is readily seen, for Du uD = u, therefore D 2 u - DuD = Du and DuD - iiD* = uD ; consequently Di + uD 2 = 2DuD + il. In like manner D 3 it- uD 3 = 2DuD + u, and the proposed result can be readily shown by induction. Hence we infer that we may write SM in the form (35). 25. We have in the next place to show that the symbolic oper ator u(A + DAiD + . . . +D"A n D"}u may be written in the form B i) + DB l D+ ... +D n B n D n . To establish this, it is sufficient to show that uD r A r D r u can be transformed into the shape in question. By Leibnitz s theorem we have D r u= ulf+ruD " + and .. +L r , But these may be written D r u = uD r + L^ 1 + LJ uD r = JD r u - D r ~ l L : + If~"L^ therefore uD r A r D r u = (D r u - D r ~ l L^ + . . .}A r (uD r + ZjZ/" 1 + . .). Hence by 24 we readily see that uD r A r D r u is of the required form. Accordingly we may write u(A + DA^ + ... + D n A n D n }uv = ( Q + DB^D + ... + D"B n D")v(BQ). If uv = 8y, we get from (35) u8M=( + D 1 D+ . . . +D n n D"fe ........ (37) ; U again, if v = l in (36), we have + ... +D n A n D n }u = B ......... (38). Hence, if u : be a solution of (A + DA 1 D+ . . . +I)"A n D")u = Q, that is, if it be a solution of 5M=0, the corresponding value of B is 0. Consequently we have from (36) 1 = D(B l + DB.JD +... If u 1 = , this becomes ~i S3f=z l D(S 1 +DB 3 D+ . . . D n ~ l B n D n ~ l )Dz l Sy ...... (39). Again, the symbolic operator B l + DB. 2 D+ . . . +D n ~ 1 B n - l D"~ 1 can be transformed into z. 2 D(C 1 + DCJ) + . . . + D n ~ C n ^D n ~~}D~.,, and so on. Finally we get 5J/ = ^Dz.,D . . . z^DsnDzn-i . . . DuDz^y ......... (40). Thus we obtain 8~U= I 1 8y 1 Dz. i D . . . Dz H D . . .D^Dz^ydx ......... (41 ). J .r If we integrate by parts, then, since the limiting values of y, y . . . are fixed, we get 5 2 7= - I Dz 1 $y)(z i Dz s . . . z.J}z^y}d,r, and after J -< o n successive integrations 8 U 17= ( - 1)" z n (Dz^. l Dz n . z D^5>j)lM ......... (42). J x-o n d~V Again, from (33) we see that A n =(-l) . .. ; also from (40) we dijW" find without difficulty 4, = *,fa*. ..*,-!) .................... (43). Hence we get finally dPf where Q n p-r- From this it follows that the sign of 5-U de- dy dV pends in general on that of Q or . Accordingly for a maxi- dyW- mum or minimum solution it is necessary that Q n should have the same sign for all values of x between the limits of integration, da: being supposed always positive. The reader will find no difficulty in applying this criterion to any of the examples which we have, hitherto given. 26. A new and complete discussion of the criteria for the discrimination of maximum and minimum solutions has been given by Mr Culverwell in Trans. Roy. Soc., vol. clxxviii. (1887). Owing to want of space we can only make an allusion to this remarkable memoir, which contains an elementary investigation of the criteria for maxima and minima not only in the case of one but for any number of dependent variables, as also for multiple integrals. 27. We now proceed to the application of the calculus of variations to multiple integrals, commencing with the Vdxdy, where V is a function of s = :r~Ti t~~j ~>i i 1 accordance with the ordinary notation. ri/ m Let Fi= / Vdy, then U= / J^dx ; therefore by (6) we have J .2/0 J *9 r x i x dU= dP r 1 dx + / l,U J x a / x r x i nn m /?/i f rvi = / 8Vdxdy+ / F8ydx + / dx ldy. J W Ho J *</ Vo x J i/o
Again 5 V- N8z + PSp + QSq + lidr + SSs + T8t, where A r = d -J~
