915 



formula (24) (mentioned again in tiie beginning of tiie present com- 

 munication), the functions a,n . The difference from the course fol- 

 lowed in the previons communication consists therefore in the determi- 

 nation of T^T^ for the particular function x^ instead of at once 

 for the arbitrary function u. This can but lead to simplification. 



We retain all notations and suppositions of N". 24, and thns 

 especially assume the existence of three numbers «, y, ;:?, having 

 the properties explained there. To begin with, we observe that x^ 

 belongs to the circle {^), hence T\{,i^) to (y), hence T^T,{x^) to («). 

 In other words '^k 'S a function that is regular in the closed domain 

 (a), and we at once add the remark that the regularity of a„i follows 

 from this by means of (24). We further develop T^{x^), as T^u in 

 N*. 24, in the series of Mac-Laukin, which, however, here simply 

 becomes the finite series (23) (copied in the previous paragraph). 

 Thus the transmutation 7, may be without any addition applied 

 term by term to that series, whereas the same operation in N°. 24 

 wanted some further explanation the series in question being there 

 infinite. We therefore have, in tei'ms with proper meanings, 



k 







valid in («). The quantity l\(a;^—^X{) may in this domain be deter- 

 mined by means of (69), since ?^i{x) as well as t»'*^-* belong to (y); 

 this gives 



7\{.vk-iXi) = {.v\-ii)'^-iXi{.v + fi). 



Substituting this result in the foregoing formula we find 



k ■ 

 , èjc = ^k{x^ii)k-iXi{.t-^li), (70) 







without anything wanting to be proved, provided we replace each of 

 the k -\- 1 terms of this series separately by its own proper meaning, 

 and add them after this being done. The proper meaning in question is : 

 substitute for the expressions (i' + ii)^~^ and ^^{x f ft) their power- 

 series in fi, multiply those series term by term, and finally replace 

 the exponents of (i by indices: then, the so obtained aggregate 

 converges absolutely and uniformly in (ft). But the same holds for 

 each new aggregate that arises from the collection of a. finite nnmber 

 of suchlike aggregates. Thus the ^ + 1 aggregates corresponding to 

 the right-hand member of (70) need not be kept apart from one another. 

 One method of grouping the elements of the aggregate consists in 

 taking all those elements with the same index of {i, or, if indices 

 have not yet been substituted for exponents, with the same exponent, 



