899 



fore for the majorant-function u of u, belonging to (q) as well as 

 u does, a transmuted 



P, (u) = ^ 



IT' 



which is regular in the closed domain {q'), while the series converges 

 uniformly in that domain. From this it follows in the first place 

 that the transmutation P, produces for the function F {u) a trans- 

 muted that is regular in the closed domain («), and secondly, in 

 connection with the normality of F, just mentioned, that this 

 transmuted is obtained in that domain, by applying the transmuta- 

 tion term by term to the series found. 

 We arrive at 



p,p,0') = yx-7^ 



~lcT~ 



(55) 



Finally, on similar grounds as in the case of formula (55), each 

 term of the last sum may be developed in (he functional series of 



Taylor. If in this case in the product }.ku{^),u{k) is considered as 

 "origin" and Xk as "increase" we arrive at the following scheme 



P.P. ('0 = 



f 



p, (») ;.„ + P\u) Z);.„ + ?y^ D-^ X, + 



2/ 



AK)^, + P\{u)DK + ^^^ ^'^. + • • •] 



2/ 



(56) 



1 



• _ _ _ P"(iii^)) - 



p,(uW);i^.+ P',(«(fc))P>Ajfc-h — TT- ^' h -\ 



Of 



in which we conceive again the P-series to be written at full, so 

 that it is a triple scheme corresponding term by term to the triple 

 scheme (56). 



If in this majorant-scheme the value x^ -{- «, is given to x, so 

 that X — x^ is real and positive, it is clear that all the terms are 

 real and positive. But we know one method of grouping these terms, 

 in which we arrive at a convergent series, viz. the one that origi- 

 nally led to the scheme. According to a well-known proposition we 

 may conclude from this that any combination of the elements into 

 one or more simply infinite series or into a simple series of such- 

 like series, regularly leads to the same sum. For other values of x 



