546 Mr. stokes, on THE CRITICAL VALUES OF 



This mode of treating the subject however appears very inconvenient, except in the case in which 

 the series are only temporarily divergent, being rendered convergent again by new integrations; 

 and even then it requires great caution. The series in question may however be rendered con- 

 vergent by tneans of transformations to which I now proceed, and which, with their applications, 

 form the principal object of this paper. 



The most important case to consider is that in which /(a;) and its derivatives are continuous, 

 so that the divergency arises from what takes place at the limits and a, I shall suppose then, for 

 the present, that /(.r) and its derivatives of the orders considered are continuous, except the last, 

 which will only appear under the sign of integration, and which may be discontinuous 



Consider first the series of sines. Suppose that f(x) is not given in finite terms, but only by 

 its developement 



f(x) = 2A sin -— , (24), 



where A„ is supposed to be given, and where the developement of /(a) is supposed to be that which 

 would result from the formula (3). I shall call the expansions of /(.t) which are obtained, or which 

 are to be looked on as obtained from the formula; (3) and (22) direct expansions, as distinguished 

 from other expansions which mav be obtained by differentiation, and wliich, being divergent, cannot 

 be directly employed. Let us consider first the even differential coefficients of f{~v), and let A'^, 



.4 J ... be the coefficients of sin in the direct expansions of f {x), f^{a;) ... The coefficient of 



sin in the series which would be obtained by differentiating twice the several terms in the 



a 



series in (24) would be A„. Now 



riTrx , 



dx : 



■^n= - / /(« ) sin - 

 a •/(, a 



and we have by integrating by parts 



2«=7r' p , . n-KX 2mr .^ . nw.v 2 . n-RX 



-S- Jf(^) sin —I-d'V =— ^/(a;)cos— -/(a;) sin — - 



a a a a a a 



+ — / / (j? ) sin ax . 



a • a 



Taking now the limits, remembering the expression for A", and transposing, we get 



<= '-^{/(O) -(-!)'■/(«)} -^ A (25). 



Any even differential coefficient may be treated in the same way. We thus get, n being even, 



+ (- i)r ? .^ |_^-»(o) - (- i)»/''-^(a)}. ... (26). 

 a a 



13. In the applications of these equations which I have principally in view, /(O), f(a), f"(0)... 

 are given, and Ai, A.;,, A^... are indeterminate coefficients. If however A,, A^ ... A„ .•• are given, 

 and /(O), f{a) ... unknown, we must first find /(O), /(o) ..., and then we shall be able to sub- 

 stitute in (25) and (20). This may be effected in the following manner. 



