SECT. IV.] WHAT TERMS MUST BE INCLUDED. 443 



In the second member we must give to i all its values, that is to 

 say we must successively substitute for ^, all the roots p, of the 

 equation (/). The integral must be taken for a from a = to 

 a = X, which makes the unknown a disappear. The same is the 

 case with /3, which enters into the denominator in such a manner 

 that the term sin p.x is multiplied by a coefficient a. whose value 

 depends only on X and on the index i. The symbol S denotes 

 that after having given to i its different values, we must write 

 down the sum of all the terms. 



The integration then offers a very simple means of determining 

 the coefficients directly; but we must examine attentively the 

 origin of this process, which gives rise to the following remarks. 



1st. If in equation (e) we had omitted to write down part of 

 the terms, for example, all those in which the index is an even 

 number, we should still find, on multiplying the equation by 

 dx sin fj,.x, and integrating from x = to x = X, the same value of 

 a n which has been already determined, and we should thus form 

 an equation which would not be true ; for it would contain only 

 part of the terms of the general equation, namely, those whose 

 index is odd. 



2nd. The complete equation (e) which we obtain, after having 

 determined the coefficients, and which does not differ from the 

 equation referred to (Art. 291) in which we might make =0 and 

 v =/(#), is such that if we give to x any value included between 

 and X, the two members are necessarily equal; but we cannot 

 conclude, as we have remarked, that this equality would hold, if 

 choosing for the first member xF (x) a function subject to a con 

 tinuous law, such as sin x or cos x, we were to give to x a value 

 not included between and X. In general the resulting equation 

 (e) ought to be applied to values of x, included between and ^Y. 

 Now the process which determines the coefficient a t does not 

 explain why all the roots ^ must enter into equation (e), nor 

 why this equation refers solely to values of a:, included between 

 and X. 



To answer these questions clearly, it is sufficient to revert to 

 the principles which serve as the foundation of our analysis. 



We divide the interval X into an infinite number n of parts 



