1865.] Mr. W. H. L. Russell on the Summation of Series. 333 



JP +fdv'V' 1 V' a: 



&c. = &c., 



the number of each set of quantities u lt u', &c., lf v', &c., M^, 10', &c. 

 being of course finite. 



I shall now consider the series 



+W)* W + &c - 

 (a;) + &c., 



where <p(x) and i//(a:) are rational functions of (#). 

 Let 



Hence, by what has been said, the problem is reduced to finding a definite 



i 



integral of the formfdu U^Ug equivalent to j , l m nx t 



Now 



+n , == _i _ -tf+jm VioglV-" 



r /_J_\Jo V "^ 



lw + wa?J 



n fW.^'Vl qg.1^" 1 

 JJ o \ &e ^ 



m + nx 



* cos 



V2 



C* * f* 



= _ j ds log,"* - . s m+ I dz cos (m + nar)^ 2 cos 2 V^r . 



,-p j __ , / 



* Jj e * Jo 



V IT .Z 



2 B 2 



