930 



liere a,' is not omIj a symbolic but a real /'•' power, viz. of the 

 number to — ,/;, which is independent of i. In consequence of this 

 the proper meaning of the pro(hict of two symbolic powers is in 

 the present case ecpial to the j)rodMcl of their proper meanings and 

 this has the effect that a^' may be replaced by (t^,i before performing 

 the involution '). The result is 



«//,»! 



(o) — a*)' tu(')(j;) 



a 







The infinite series within the l)rackets (without (he term — x) is 

 clearly a formal expansion of the expression 



and if N'. 20 of the preceding communication is consulted, especi- 

 ally formula (60), it will l)e evident that the present expansion does 

 represent the last mentioned quantity in the domain («J. Finally 

 we have therefore 



«;/,.« = !('-> [<'-» (^)l — .'?'•]"' , 

 which of course could be deiived in a much simpler manner (obser- 

 ving that S,,>So> is itself an operation of substitution in which the 

 function [to[a>(.i')] has to be substituted for x). 



In the second place we take 1\ = Sr.„ 7', = D—K Here a^,i is 

 the same as in the preceding case and 



as we have already utilized several times (see for instance N°. 16, 

 3'*^ communication). Since the first component is the same as in the 

 first case, formula (91) will apply here as well. But the order of 

 the involution and the replacement of aj by a^,! must not be 

 inverted now, the symbol a^^ not being the z^'> power of a number 

 independent of /. However, an other reduction is permitted. For we 

 also have 



X 



a2,j :=r I (< — a)' dt 







so that a^,i is at least the integral of an i"^'' power. We may there- 

 fore give as a further rule for the reduction of the power-series in 

 the right-hand member of (91) the following one: replace a^^ by 



^) It is for the same reason that an exact result is obtained if in applying 

 formula (24), that is am = (S — x)>'> ^ to the operation of substitution, ^ is replaced 

 by u before the expansion of the binomial. 



