I 



THE SUMS OF PERIODIC SERIES. 535 



SECTION I. 



3Iode of ascertaining the nature of the cVisconthmity of a function which is expanded 

 in a series of sines or cosines, and of obtaining the develupements of the 

 derived functions. 



1. By the term function I understand in tliis paper a quantity whose value depends in any 

 manner on the value of tiie variable, or on the values of the several variables of which it is com- 

 posed. Thus the functions considered need not be such as admit of being expressed by any 

 combination of algebraical symbols, even between limits of the variables ever so close. I shall 

 assume the ordinary rules of the differential and integral calculus as applicable to such functions, 

 supposing those rules to have been established by the method of limits, wliich does not in the least 

 require the possibility of the algebraical expression of the functions considered. 



The term discontinuous, as applied to a function of a single variable, has been used in two 

 totally different senses. Sometimes a function is called discontinuous when its algebraical expression 

 for values of the variable lying between certain limits is different from its algebraical expression for 

 values of the variable lying between other limits. Sometimes a function of ,r, /(.r), is called con- 

 tinuous when, for all values oi x, the difference between /(.r) and/(.t'±/i) can be made smaller 

 than any assignable quantity by sufficiently diminishing A, and in the contrary case discontinuous. 

 If /(.r) can become infinite for a finite value of x, it will be convenient to consider it as dis- 

 continuous according to the second definition. It is easy to see that a function may be discon- 

 tinuous in the first sense and continuous in the second, and vice versa. The second is the sense in 

 which the term discontitiuous is I believe generally employed in treatises on the differential calculus 

 which proceed according to the method of limits, and is the sense in which I shall use the term in 

 this paper. The terms continuous and discontinuous might be applied in either of the above senses 

 to functions of two or more independent variables. If I have occasion to employ them as applied 

 to such a function, I .shall employ them in the second sense ; but for the present I shall consider 

 only functions of one independent variable. 



In the case of the functions considered in this paper, the value of the variable is usually sup- 

 posed to be restricted to lie within certain limits, as will presently appear. I exclude from 

 consideration all functions which either become infinite themselves, or have any of their differential 

 coefficients of the orders considered becoming infinite, within the limits of the variable within whicii 

 the function is considered, or at the limits themselves, except when the contrary is expressly stated. 

 Thus in an investigation into which /(a) and its first n differential coefficients enter, and in which 

 f(v) is considered between the limits tv = and .r = a, those functions are excluded, at least at first, 

 which are such that any one of the quantities /(.r), f'{w) ... f"(.v) is infinite for a value of .v 

 lying between and ri, or for x = or <v = a ; but the differential coefficients of the higher orders 

 may become infinite. The quantities f{x), f{x) ...f°(x) may however alter discontinuously 

 between the limits x = and x = a, but I exclude from consideration all functions which are such 

 that any one of the above quantities alters discontinuously an infinite number of times between the 

 limits within which x is supposed to lie. 



The terms convergent and divergent, as applied to infinite series, will be used in this ])aj)cr in 

 their usual sense ; that is to say, a series will be called convergent when the sum to n terms 

 approaches a finite and unique limit as n increases beyond all limit, and divergent in the contrary case. 

 Series such as 1 - 1 -f I - ..., sin x + sin S x + sin 3 x + ..., (wluro x is su])posed not to be or a 

 multiple of TT,) will come under tlie class divergent; for, although the sum Ion terms does not 

 increase beyond all limit, it does not approacii a unique limit as /i increases beyond all limit. Of 



3 7.2 



