918 



we have met with anotlier, viz (71), which expresses the resulting 



quantities §,„ also in X,„ and .u,,,. But if we want this formula as a 



final result we had better write it in the following simpler form 



$,„ = (A -f.r)'« ={(;. + . c)'"},..^,,,. ..... (74) 



32. Formulae (73) and (74) are valid in the domain («), as it 

 has been shewn in the foregoing paragraph. It still remains to 

 be proved by means of (73) that the resulting series P is complete 

 in («) with a corresponding domain that is at most equal to [^), a 

 statement we gave in the previous communication. To do this we 

 shall make use of the following proposition, the proof of which we 

 do not give, first because it is very easy, and secondly because the 

 proposition may perhaps be established elsewhere: 



The upper limit for m = cc 



1 

 Urn I F,n -h Q,n I '« , 



HI = 00 



of the 7)1^'' root of the modulus of the sum of two complex quantities 

 Pm (ifid Qm , both defined in the aggregate of positive integral m- 

 values, is equal to the greatest of the two upper limits 



1 1 



Urn I P,„ I '" , lim. I Q„, | '« 



?n = 00 7/1 = 00 



of the 7n''' roots of the moduli of those tioo quantities separately. If 

 the two latter limits be equal then the former is never greater than 

 each of them. 



An analogous proposition is, as a corollary of the one just men- 

 tioned, valid for a sum consisting of an arbitrary finite number of 

 terms, this number not depending on m. 



The proposition will serve us to investigate the m^^ root of the 

 modulus of the coefficient a„, (x) of the resulting series P. If we 

 work out the right hand member of (73) in the prescribed manner, 

 we obtain 



V-- r , f* , /^' 1 , 1 



= yk 



mk 



Zl — a J' ^^^> 







this equality containing only proper expressions. We assume again, 

 as in the previous communication, that (y) is not the maximum domain of 

 completeness for the series P^, so that there is a domain (y') ^ (y), 



