the Calculus of Variations. 325 



less than oo / — co ll we must have ju. dx / -f v 8 x n = by the com- 

 mon theory. 



This equation ju, 8 x, + 7r 8 J",, = resolves itself into two 

 others, since 8 ar y and 8 xj) are independent of one another, in 

 which x, and x tl are the only unknown quantities : they there- 

 fore may be determined by means of them. 



Now 00,—cO/j being the value of U/ — U,/ when #, + 8.1^ is sub- 

 stituted for x t and x u + ^x n for x u in it; and since x / and x n 

 enter into U, and U /y respectively, partly in consequence of 

 their being involved in U, and partly in consequence of the 

 change of x into x t and x a respectively wherever x occurs 

 in U; it is evident that w ; may be obtained by substituting 

 Xj + 8 Xj for x / and x n -f 8 x u for a? // wherever they occur in 

 U, and then substituting x + 8 x for x, and afterwards chang- 

 ing x and 8x into x t l and 8 a;, in the result. In like man- 

 ner we may find a> //} or the value, of U,, when x / + dx / and 

 x lt + 8 a7 ;/ are substituted for x t and a?^ in it, by first substitu- 

 ting x / +lx / and x /t -\- lx Jt for x t and a;,, wherever they occur 

 in U, and then substituting x + 8x for x, and afterwards 

 changing x and 8 a? into x n and lx /t . All these substitutions 

 are to be made not only where x / and x u occur alone, but 

 likewise where they occur involved in the functions y t and y n . 

 In like manner since the operation of substituting x + 8 x for 

 x in U is equivalent to the substitution of x / + 8 a, for x /9 

 when a? / is obtained ; and equivalent to the substitution of 

 x n + lx u for x JP when <ojj is sought; it is evident that in one 

 case y must be considered as the same function of x that y t 

 is of x)\ and in the other y must be considered the same 

 function of x that y n is of x u . 



Since we only want those terms which are multiplied by the 

 first powers of 8 a;, and \x lP we will carry the operations in 

 what follows only to that extent. We will also use 83/, to 

 denote the term multiplied by 8 a:, in the new value of 3/, ob- 

 tained by substituting a;, + 8 a;, for x, in it. In like manner 

 ly n denotes the term multiplied by 8 a;,, in the new value of 

 y u when a*,, + 8 a;,, is substituted for x u in y n \ and ly denotes 

 the corresponding term when a; + 8 a; is substituted, for x my 

 considered as a function of x : Ip the term in p : and so on. 



Now, if U contain x / y / x n y lfl and we substitute x / -f 8 x / 

 for a:, and a;„-f 8a;„ for a;,,, the new value of U thus obtained is 



U + -^ 6x < + -^^<+ Z^+^/,+ &C. 



but U ■= a +fdxY, and a may be considered either as a 

 function of x t y t or of x u y n \ we will suppose « to be a 



function 



