SECT. IV.] SERIES EXPRESSED BY DEFINITE INTEGRALS. 425 



414. We may employ very different processes of investigation 

 to express, by definite integrals, the sums of series which repre 

 sent the integrals of differential equations. The form of these 

 expressions depends also on the limits of the definite integrals. 

 We will cite a single example of this investigation, recalling the 

 result of Art. 311. If in the equation which terminates that 

 Article we write x + 1 sin u under the sign of the function c, 

 we have 



1 l&quot;du &amp;lt;j&amp;gt;(x + t sin u) - + (*);+ a &amp;lt;/&amp;gt;&quot; (x) + =Ai ^ (.r) 



7T .. o 4 .* 



Denoting by v the sum of the series which forms the second 

 member, we see that, to make one of the factors 2 2 , 4 2 , 6 2 , &c. 

 disappear in each term, we must differentiate once with respect 

 to t, multiply the result by t, and differentiate a second time with 

 respect to t. We conclude from this that v satisfies the partial 

 differential equation 



d~v _l d^f dv\ d^v_(Fv Idv 



dx* ~ 1 It ( t ~dt) cU?~~d? +: tdt 



We have therefore, to express the integral of this equation, 



1 [ n 

 v = I du (j&amp;gt; (x + 1 sin 11) + W. 



The second part W of the integral contains a new arbitrary 

 function. 



The form of this second part W of the integral differs very 

 much from that of the first, and may also be expressed by definite 

 integrals. The results, which are obtained by means of definite 

 integrals, vary according to the processes of investigation by which 

 they are derived, and according to the limits of the integrals. 



415. It is necessary to examine carefully the nature of the 

 general propositions which serve to transform arbitrary functions : 

 for the use of these theorems is very extensive, and w r e derive 

 from them directly the solution of several important physical 

 problems, which could be treated by no other method. The 



