CuLVERWELL — Solutions in the Calculus of Variations. 387 



are only permitted to be small. Then, remembering tliat we cannot 

 apply Taylor's expansion to the A variations except when they are 

 small, the complete difference between the original and varied integral 

 may be written 



1+6) - ^= -^(1 + 6 



M + 



i^(i,S)<?o-- Fd% 



\^ Fda- 



= hFd% + (h) + (^(i+A) -F+ SFa^A))(l(T + {h) da-, 



where {h) indicates quadratic terms. 



But since the first variation must vanish always, we have 



■'S. + a 



SFds = 0, 



{" SFd:^+ rAFd(T+ [ 



SFd:^ + AFda- + SFda- = 0. 



Fag. 5. 



Subtracting this from the expression just written, 

 /(iH-S) - /= ["(-^(i+A) - F- AF) da + ["(8i^(i,A^ - hF) d<T + {l)„ 



where (^)2 is written for small terms of the second or higher orders. 

 Hence 



h.,-I- 



Edd + 



^ {Jc\da\{l\ 



Fdcr, 



neglecting small terms, because evidently the tenn Ai^ is the term 

 which appears in E. For, in the first place, that the solution is 



