6 REPORT 1859. 



to render it more accordant with the symbols of integration with which it is involved 

 in researches of this kind. 



My sole object in recalling these well-known facts has been to indicate the 

 starting-point of my own researches on the calculus of variations, a few of whose 

 results I have now the honour to communicate. 



Following the example of M. Sarrus, I apply the term definite expression to any 

 function whatever in which fixed values have been substituted for all the indepen- 

 dent variables. Such a definite expression is no longer a function of these variables ; 

 it depends solely upon the parameters, that is to say upon the indeterminate con- 

 stants, which it happens to contain. 



It is at once seen that every definite integral possesses this property, the variables 

 themselves being therein replaced by certain limiting values. In order to indicate, 

 in other expressions, that a certain variable must be replaced by a particular value, we 

 shall employ the symbol | ; so that 



u, or simply 



denotes the result obtained by substituting the value x\, in place of the variable x, in 

 the expression u. In conformity with the notation of the integral calculus, the 

 same symbol will also serve to denote the difference between the results obtained by 

 two different substitutions. Thus the notation 





u, or simply u 



x = x. 



denotes that the variable must be successively replaced by x 1 and « a and the first 

 result subtracted from the second ; so that 



u = 



The definition of a definite expression may now be more precisely expressed thus : 

 it is a function submitted to integrations or to substitutions with respect to each of 

 its independent variables. Such an expression is invariable so long as the constants 

 which it contains preserve the same fixed values. But if it should contain an inde- 

 terminate parameter whose value changes, the expression in question will become a 

 function of that parameter, and under this point of view may be differentiated. 



This being granted, the most general problem of the calculus of variations, the 

 problem in which, in fact, the whole theory of this calculus is contained, consists in 

 finding the derived -function of any definite expression with respect to a variable 

 parameter. M. Sarrus, it is true, has rendered the determination of this derived 

 function possible in every particular case, but neither he nor Cauchy has given a 

 general rule of differentiation applicable to every definite expression. I believe I have 

 established this rule, and in the following manner. 



Let us suppose the function V, containing any number of independent variables 



&) y> %> ■ • • • s> t, 



to be subjected to integrations or substitutions with reference, successively, to each 

 of the variables according to the inverse order of their enumeration ; so that the first 

 operation shall refer to t, the second to s, and so on up to the last, which shall have 

 reference to x. 

 Further, let 



x l ,y l ,z v .,>. s lt t t 

 be the inferior, and 



*a» Vz* ^a# • ■ • • ^2» 't 

 the superior limits of these variables. 



The limits t r and t 3 of the variable t may be functions of x, y, x, s, but they- 



