Convergence of Fourier s Series. 129 



the results obtained, reference must be made to Sachse's paper 

 already mentioned. 



10. The method employed by Dirichlet to determine the 

 value of S„ when n = oo is to break up the integral into the 

 sum of elements which are alternately positive and negative, 

 that is, into an alternating series with terms of finite magni- 

 tude. The manipulation of this series is, however, very 

 laborious, and the method of evaluating S„ by means of it is 

 long, and highly involved and indirect, and consequently is 

 not suited to the needs of the average mathematical student. 

 The investigation given in the following paper is a simplified 

 form of Dirichlet's in the sense that it depends upon the 

 evaluation of the same integral S«. But the difficulties at- 

 tending Dirichlet's evaluation are avoided by breaking up 

 the integral into three portions, two of which are of finite 

 range, the limits being — it to —h, and h to it respectively, 

 while the third portion is taken between + h 9 h being infi- 

 nitely small. It is then easy to show in a simple and 

 straightforward manner that the two first portions vanish 

 when n=x),and that, therefore, the value of the integral 

 depends only upon the infinitely thin strip taken between 

 ±h. By this means we "are enabled not only to evaluate S n 

 more easily and directly, but to investigate the limitations to 

 which the function ¥(x) must be subjected in a simpler 

 manner. For, as we shall see, the conditions that have to be 

 fulfilled by the function F(#) in order that the terms of the 

 series may be finite and determinate, and that the ?ith term 

 may be infinitely small when n= go, which are conditions 

 that have to be fulfilled in the case of every series, are sufficient 

 to ensure that the two portions of S» which lie outside the 

 limits +h vanish when 7i = co . The difficulties attending the 

 determination of the remaining conditions to be fulfilled by 

 the function are thus removed to the infinitely small portion 

 of it which lies between +h. The investigation is given, 

 first, for the case of functions which obey the laws of the 

 differential calculus, this being the only case which occurs in 

 ordinary analysis. Afterwards, the case of functions in which 

 this condition is not fulfilled is taken up. 



II. . 



11. Let F(#) be a finite, single- valued, and continuous 

 periodic function ; and where continuous, let it be differen- 

 tiable. Then, since F is periodic, and of period 2tt, the limits 

 of integration may be shifted through any distance at pleasure, 

 provided the interval between them remains unaltered and 

 equal to 27T. Hence, whatever may be the value of the sum- 



