138 Mr. W. Williams on the 



the other two. This illustrates the nature of the difficulties 

 encountered in dealing with functions of this kind, and the 



c 



danger of applying to them, without a special examination, 

 processes which have been derived only from the study of 

 functions possessing ordinary continuity. It is precisely in 

 the case of functions of this kind that the integral 



J-h 2 (/ 



becomes indeterminate in value when n= co . If the function 

 possesses ordinary continuity we know that the integral va- 

 nishes ; otherwise the integral may be quite indeterminate. 



For the infinite number of oscillations of ^—rn — — when 



n — co may conspire with the oscillations of F(# -+- 6) — F {x) to 

 produce any value whatever, finite or infinite. Incases of this 

 kind we can determine nothing as to the value of the integral 

 until we know something as to the nature of the continuity of 

 the function ; for the ordinary definition of a continuous 

 function is too general, and does not confer upon the function 

 enough properties to enable us by means of known processes 

 of integration to evaluate the integral. 



27. The conditions under which Fourier's series has been, 

 up to the present, proved to be convergent are : — 



i. That the function F(#) must not become infinite. 

 ii. It must be continuous and determinate except at a 

 finite number of points, where it may change abruptly in 

 value or experience a discontinuity. 



iii. It must, wherever it is continuous, possess ordinary 

 continuity. 



These conditions are sufficient for all the cases that occur 

 in ordinary analysis. The third condition, moreover, is 

 necessary in all such cases, since processes involving differ- 

 entiation constitute an essential part of the Infinitesimal 

 Calculus. From the point of view of the general theory of 



