XL. On the Critical Values of the Sums of Periodic Series. Bij G. G. Stokes, M.A., 

 Felloiv of Pemhrohe College, Cambridge. 



[Read December 6, 1847.] 



There are a great many problems in Heat, Electricity, Fluid Motion, &c., the solution of 

 which is effected by developing an arbitrary function, either in a series or in an integral, by 

 means of functions of known form. The first example of the systematic employment of this 

 method is to be found in Fourier's Theory of Heat. The theory of such developements has 

 since become an important branch of pure mathematics. 



Among the various series by which an arbitrary function /(.r) can be expressed within 

 certain limits, as and a, of the variable ,v, may particularly be mentioned the series which 



proceeds according to sines of — and its multiples, and that which proceeds according to 



cosines of the same angles. It has been rigorously demonstrated that an arbitrary, but finite 

 function of x, /(r), may be expanded in either of these series. The function is not restricted 

 to be continuous in the interval, that is to say, it may pass abruptly from one finite value to 

 another ; nor is either the function or its derivative restricted to vanish at the limits and a. 

 Although however the yossihility of the expansion of an arbitrary function in a series of sines, 

 for instance, when the function does not vanish at the limits and a, cannot but have been 

 contemplated, the utility of this form of expansion has hitherto, so far as I am aware, been 

 considered to depend on the actual evanescence of the function at those limits. In fact, if the 

 conditions of the problem require that /(O) and f(a) be equal to zero, it has been considered 

 that these conditions were satisfied by selecting the form of expansion referred to. The chief 

 object of the following paper is to develope the principles according to which the expansion of 

 an arbitrary function is to be treated when the conditions at the limits which determine the 

 particular form of the expansion are apparently violated ; and to shew, by examples, the advantage 

 that frequently results from the employment of the series in such cases. 



In Section I. I have begun by proving the possibility of the expansion of an arbitrary 

 function in a series of sines. Two methods have been principally employed, at least in the simpler 

 cases, in demonstrating the possibility of such expansions. One, which is that employed bv 

 Poisson, consists in considering the series as the limit of another formed from it by multiplying 

 its terms by the ascending powers of a quantity infinitely little less than I ; the other consists in 

 summing the series to « terms, that is, ex])ressing the sum by a definite integral, and then con- 

 sidering the limit to which the sum tends when n becomes infinite. The latter method certainly 

 appears the more direct, whenever the summation to n terms can be effected, which however is 

 not always the case ; but the former has this in its favour, that it is thus that the series present 

 themselves in physical problems. Tlie former is the method which I have followed, as being that 

 which I employed when I first began the following investigations, ,uid aicordinglv that which iiest 

 harmonizes with the rest of the paper. I should hardly have ventured to bring a somewhat 

 modified proof of a well-known theorem before the notice of this Society, were it not for the 

 doubts which some mathematicians seem to feel on this subject, and because there are some ])i)inls 

 which Poisson does not seem to have treated with sufficient detail. 

 Vol.. VIII. Part V. 3Z 



