456 



.(«+!) , >+!)„ L _L.f"+lL,„«-n-l^ 



+ 



+ «m C„ 



Now, it is far from certain that this amount may be rendered 

 arbitrarily small. When ?? has been fixed {n = iig > Q), an integer 

 M^Hg may be chosen such, that the part indicated by the bracket 

 is smaller than e, for m > M, but since M for this purpose has 

 perhaps to be chosen much Imager than n^, the remaining part of 

 the scheme may contain a?i important number of terms, so that 

 nothing is to be said about the total amount of them. For, though 

 the complete, that is injinite series produce an amount smaller tlian 

 2e in the horizontal rows from the ordinal number n -j- 1 down to 

 m, — as it follows from the first of the inequalities (30), — this need 

 not at all be the case with the finite parts of those series, if terms 

 of different arcjument occur in them. 



So it appears impossible to prove, even when both members exist, 

 the equality (29), and with this Booklet's theorem X in its generality 

 becomes illusory. Also Pincherle, who gives the same development 

 and calls it "série fonctionelle de Taylor", points out that the latter 

 does not hold without more, and that it has to be investigated in 

 each special case. He does not mention, however, a category of 

 cases, in which its validity maj' be stated, and from the preceding 

 investigation it is clear that this is not at all easj' : we see for instance 

 that it is not sufficient that the expression (31) for the /zr/^fra/ ???*'/ji'ö?'a?2^ 



u of the function u, that is 



M= JCJ -f |C,| .V 4- |C,| y' + • . • ,(2/ = ^ — .^o) 



may be made arbitrarily small in all the points of the domain (o), 

 in order to deduce from this that this holds as well for u itself; 

 this would only be possible if all a's were also majorant functions. 

 Though the preceding investigation leads to a negative result, yet 

 it is not without interest, as from it clearly appears, what is wanted 

 for a statement of the theorem : without a special supposition con- 

 cerning the rt's we cannot arrive at a result. As to this point, we 

 have seen that the series P is, shortly speaking, continuous, if it 

 represents a complete transmutation ^). 



1) It ought not to be supposed that if Pu exists in the domain («) for an 

 arbitrary function of F{T), the completeness of P would follow from it. This 

 would only be the case if the functional field F{T) contained all the functions 

 belonging to a certain circle (o), but it is not at all unimaginable that T is defined 

 in such a way that this is not the case. 



