THE SUMS OF PERIODIC SERIES. . 543 



limit of nA„ is finite, by the method which will be explained in the next article, the part which 



remains will be at least as convergent in the former case as the series h - ... ^ (- ..., whereas 



we cannot affirm this to be true, and in fact it may be proved that it is not true, in the case in 

 which /"(*) becomes infinite. Observing that the same remark will apply when we come to 

 consider the critical values of the differential coefficients of /(>r), I shall suppose the functions and 

 derived functions employed in each investigation not to become infinite, according to what has 

 been already stated in Art. 1. 



8. After having found the several values of a, and the corresponding values oi N — M, we 

 may subtract the expression (10) from A„, provided we subtract from the sum of the series (15) 

 the sums of the several series such as (11). Now if X be the sum of the series (ll), 



X = - (N - M) {^ - sin ^ + S - sin ^ >. ... (19). 



TT I w a n a ] 



But it has been already shown that 2 — sin nz = kiir-x) when z lies between and Stt, =0 



n ^ 



when z = 0, and = - ^ (tt + «) when « lies between and - 2 tt. Now when .r lies between and a. 



TT {x + a) ,. , , . TT {as - a) ,. , , , , ,. , 



lies between and 2 tt, and lies between - 2 tt and ; and when x lies between 



a a 



, TT (;(? + a) ... ... , , TT (x — a) ,. , , ... 



a and a, still lies between and 2 tt, and — ^^ now lies between the same limits. 



Hence 



,7' 



X = - {N - M) - , when x lies between and a 

 a 



= {N — M) , when x lies between a and a 



(20). 



We need not trouble ourselves with the singular values of the sum of the series (15), since we 

 have seen that a singular value is always the arithmetic mean of the values of the sum for values 

 of X immediately above and below the critical value. This rule will apply to the extreme cases in 

 which J? = and x = a, if we consider the sum of the series for values of x lying beyond those 

 limits. The rule applies to the series in (19), which is only a particular case of (15), and con- 

 sequently will apply to any combination of series having this property, formed by way of addition 

 or subtraction ; since, when we increase or diminish any two quantities jI/„, N„ by any other two 

 M, N respectively, we increase or diminish the arithmetic mean of the two former by the arithmetic 

 mean of the two latter. 



It has been already stated that we may, with a certain convention, include quantities referring 

 to the limits x = and x = a under the sign of summation S. If we do so, and put H for the 

 sum of the series {\r>), and /?,. for the remainder arising from subtracting the expression (10) 

 from A„, we shall have 



S - SX = ^B„ sin , 



a 



1 and the sum of the series forming the right-hand side of this equation will be a continuous function 

 'of iT. As to SX, the value of each series contained in it is given by ecjuation (20). 



To illustrate this, suppose H the ordinate of a curve of which x is the abscisisa. Let OG be 

 the axis of x ; OA, MB, ND, Gb right lines perpendicular to it, and let OG = «. Let the curve 



4 A 2 



