126 Mr. W. Williams on the 



afterwards see, even when we limit ourselves to functions 

 which are finite and continuous, the limitation is too general, 

 and we cannot determine whether Fourier's series is con- 

 vergent or not until we know something of the nature of the 

 continuity of the function. 



2. The object of the present paper is to simplify the 

 investigation of the subject, to bring it within the reach of 

 the student acquainted only with the elements of the Infini- 

 tesimal Calculus, and to exhibit in an elementary manner the 

 nature of the difficulties that have to be surmounted and the 

 principal results obtained. At the same time, in addition to 

 simplifying the discussion, and rendering it perhaps more 

 interesting and instructive, it is hoped that some additional 

 light will have been thrown upon the question of the con- 

 vergency, and that the limits within which the convergency 

 holds will be found to be to some extent widened and more 

 clearly discussed. 



3. The literature of Fourier's series is very extensive, few 

 mathematical subjects having, perhaps, been so widely dis- 

 cussed. A very valuable account, both critical and historical, 

 of the chief investigations into the subject has been given by 

 Arnold Sachse (" Yersuch einer Geschichte der Darstellung 

 willkuhrlicher Functionen einer Variabeln durch trigonome- 

 trischen Reihen," Gottingen, 1879) in an essay which has been 

 translated and published in the Bulletin des Sciences MathS- 

 matiques, vol. iv. (1880). It is not proposed to enter here 

 into the history of the subject, or to discuss the elementary 

 properties of Fourier's series, such properties being treated 

 and illustrated in ordinary text-books. We have here to take 

 Fourier's series in its most general form, as it stands, and 

 determine the conditions under which it is convergent. 



4. Fourier showed that if an arbitrary function of x can 

 be expanded into a series of the form 



¥(x) = ^a + %a n cos nx + %b n sin nx, 

 i i 



the coefficients will be determined by the definite integrals 



a n = -\ F(v)cosnvdv, b n =—\ ~F(v) smnv'ftv, 



v being written for x under the sign of integration. To 

 investigate the possibility of the expansion, it is, therefore, 

 necessary to determine the most general conditions which the 

 function must satisfy in order that the series thus defined may 

 be convergent and tend to the limit F(^). 



5. Of the different methods that have been employed in 



