929 



wanted to effect the final operation of tlie given rule on the infinite 

 series which we have obtained for ^7/,,,, {,v), and not on this function 

 considered as a whole. But that the second interpretation will do as 

 well, follows again from the consideration of the natural majorants 



35. In applying the preceding symbolic formulae to particular 

 cases special reductions are often necessary which have to be justified 

 individually. Only in order to call the attention to this point we shall 

 discuss one or two examples, but for the rest further explanations 

 by means of examples of the symbolic formulae may be omitted 

 after the detailed consideration of examples with the transmutations 

 I)—^ and So, in the previous communication, the more so as the 

 formulae in question provide the same genei^al results relative to 

 resultant magnitudes as the formula of Bourlet. 



We first take the case T^ = T,=^ So,- Here 



«i,t = a2, J- = (to — .?;)'■. 



Applying (73) we successively obtain 



m m 



ail,m = \l ni/c a,"'-^ai,A; G^* + a^) =z\k mjc a,'"-^' [iü{x +(<,) — (.'• + a^)]/^ 







= [to {x + a,) — .-r]'». 

 Here the reduction of the last member but one to the last is 

 allowed in consequence of the particular form of the given trans- 

 mutation. Nevertheless the exactness of the reduction is included in 

 the general considerations according to which analytic reductions are 

 permitted. The resulting expression must now be expanded in a 

 power-series of a^, which must really exist, according to the general 

 theory, if x lies in the domain (t^J ; this is in fact the case since 

 the circle of convergence of to is greater than a domain of com- 

 pleteness («1) of aSco and thus a fortiori greater than («,). We may 

 also obtain the power-series in question by raising that for m = 1 

 to the m'^^ power, according to the common rule for the involution 

 of an infinite series; this is meant, when we write 



«//,m = 



"Jl. a,«to(0(.r) 



(91) 







where the term — x has to be combined with the term corresponding 

 to e=:0 under the sign of summation. 



When the involution has been performed we have to replace a\ 

 by a,, J. The peculiar thing to be noticed is now that we may invert 

 the order of the last two operations; this is caused by the fact that 



