931 



{i — x)i and integrate evei-y term froni / ;= O to ^ = x. This repiace- 

 nment maj now be performed before the involution, on the same 

 ground as above, so that w^e may say : the power-series in {t — x) 



^ a 







(92) 



where the term — x must be taken together with the term under 

 the sign of summation that corresponds to i = 0, must be involved 

 to the m"^*^ power according to the common rule for the involution 

 of infinite series, and the result is to be integrated term by term 

 between the limits ^ = and t = x. Now the infinite series in (91) 

 (without the term — x) is a formal expansion of to(^) and if N°. 28 

 of the preceding communication is consulted, from which it appears 

 that the radius of convergence of mix) is greater than 2aj, we infer 

 that the expansion is essential for values of x in the domain {a^). 

 The involution to the m^^ power therefore leads to a power-series 

 in {t — x) which, for the ^-values mentioned, represents the function 

 \oi{t) — ^]"' and since the convergence of the latter power-series is 

 uniform in the integration-interval, its integration term by term 

 produces the integral of the function represented by it. Thus we 

 finally have 



ƒ 



«77, m= I [t^(0 — ^Ydt 

 



which has also been found in N" 28. 



In applying formula (88), in order to determine sn,/n. analogous 

 peculiarities occur in either of the cases just mentioned. 



36. In the two previous communications we have, in considering the 

 resultant of two complete transmutations, been able to simplify our 

 statements by supposing that we had to deal with the following case. If the 



functions a„j(a?) or shortly a^a be the natural majorants of the coef- 

 ficients a„i of a series Prepresenting a transmutation which is complete 

 in certain circular domains («), with common centre x,^ and a radius 

 « varying between and a certain positive value A, then the 



'maximum value a{a), for the domain («), of the upper limit 



1 



ax^li'm\am\'"^ (6') 



is luithin the interval (0, A) a continuous function of («)• (Cf. N". 23j. 

 From this it might further be derived that the corresponding quantity 

 (i{ii), belonging to the given functions am themselves, was equal to 



