and on Go-resolvents. 

 III. It foUows that 



7)?/ 



[q^Y ^r = 



V a a 



-r 



\vn 

 a 



a 



] —n 



But 



[ r n ^ m ^-, , ^^ ir-a-i ^ [ f^, 



Ir-a-l 



193 



(h) 



J 



[ r n 



m 



a 



(1 + n) 



Ir-a-l + n 





--y' n on 

 a a 



\ T n m 

 I a a 



In-a 



J 



J 



\ rn ni ]" 



[ a a J 



(i) 



and (h) becomes 



[rfitr= - 



\ rn m Y 



[ a a J 



[ 9' n ni 



a 



J 



«,-, 



(J) 



IV. Proceeding as in the former pait of this paper we, 

 slightly changing the form of (j), infer that, since 



[r]« 



Cn — a) r m 

 a a 



Ur 



r n , 7)1 

 . a a 



1 



"l^r-a = (k) 



consequently that, as in a former case, the co-resolvent will 

 be 



m 



(n — a) D . ni 



u 



nD , m -.1" „ ^ n^ 



+ 1 x^'u = (1) 



a a i ^ ^ 



in which last equation n represents y"\ This (1) is the co- 

 resolvent of (a) for the case of 2^1= 1. We may obtain it 

 for any value of by multiplying the last term of (1) into 

 z"''", for the exponent of z will be 



r n , m ( ir — a) n , m 



— + r — ^ '— + — 



a a ^ a a 



r + a 





n 



We shall thus give to the co-resolvent an extension equiva- 

 lent to that which Mr. Harley has sought to give to another 



o 



