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PROCEEDINGS 



CAMBRIDGE PHILOSOPHICAL SOCIETY. 



December 6, 1847. 



On the Critical Values of the sums of Periodic Series. By G. 

 G. Stokes, M.A., Fellow of Pembroke College, Cambridge. 



There are a great many problems in heat, fluid motion, &c, the 

 solution of which requires the development of an arbitrary function 

 of x,f{x), between certain limits as o and a of x, by means of func- 

 tions of known form. The form of the expansion is determined, at 

 least in part, by the conditions to be satisfied at the limits ; and it 

 is usually considered that these conditions are satisfied by adopting 

 the form of expansion to which they lead. Thus, if the problem 

 requires that/(o) and /(a) vanish, it is considered that this condition 



is satisfied by developing/^) in a series of sines of — and its mul- 



a 

 tiples. But since an arbitrary function admits of expansion in such 

 a series, the expanded function is not restricted to vanish at the 

 limits o and a. It becomes then a question, how shall we know 

 when the expanded function does really vanish at the limits, and if 

 it does not, how are such expansions to be treated, and are they of 

 any practical importance ? 



In considering the logic of such developments, the author was led 

 to perceive in what manner the evanescence of /(#) at the limits can 

 be ascertained, or else the values of/(o) and /(a) obtained, from the 

 development itself, even when the series cannot be summed, by ex- 

 amining the coefficient of sin in the nth. term. In a similar man- 

 es 



ner the discontinuity of/(<2?) or any of its derivatives may be ascer- 

 tained, and the amount of the sudden change of the function deter- 

 mined. In such cases the expansions of the derivatives off(x) can- 

 not be obtained by differentiating under the sign of summation, but 

 are given by formulae which the author has considered. 



The most important case in considering a series of sines, is that 

 in which f{x) is continuous ; but /(o) and/(«), instead of being equal 

 to zero, are given quantities, and the coefficients in the expansion 

 are indeterminate. In this case the coefficients in the expansions of 

 f'(x) and f"{x) contain, in addition to the indeterminate coefficients 

 which enter into the expansion oif(x), the given quantities /(o) and 



No. V — Proceedings of the Cambridge Phil. Soo. 



I9C 



