610 



30 



T f "STm Um 1 <^ T, if n> N. 



Now we have, on account of the additive property of 2\ in the 

 numerical field of operation 



T u = T u^ -\- T u^ + . . . -|- T u„^i + 7' I \m u,n j, 



n 



from which it follows in connection with the result just arrived at, 

 that we also have in that field 



CO 

 7' II =: \m T Ui,i , , 







the series converging uniformly there. Thus the proposition has 

 been proved. 



If 7* is a normal transmutation both the N.F.F. and the N.F.O. 

 are a circle with centre x\. The radii of these two we call respect- 

 ively {(i) and («). The characteristic expansion of u is in that case 

 the one in a power series 



Then, if we call §',„ the function in which {x — .v)>" is transformed 

 by T (the latter e.ci.'its and is regular in (a) since the rational integral 

 functions form part of the F.F. of a normal transmutation), we 

 have in («) 



7' (u) =: r^^\ 4- c'j,i''j + ...-)- C,n^'m -f • • • ^ 



and the series converges there uniformly. This is the proposition 

 we referred to and which we may express as follows: 



If T he a n r m a I additive transmutation, of ujhich the circle 

 {(i) forms the N.F.O., and the circle [c>) the \ .F.F. tlie transmuted Tu 

 of a function u of the F.F. may be found by applying the trans- 

 mutation term by term to the pover series, in ivhich u may he deve- 

 loped in the domain {a), and the result is a series that converges 

 uniformly loithin («). 



19. We shall now discuss the generalisation of Taylor's theorem 

 for the functional calculus. 



This is concerned with the development of the transmutation T, 

 applied to a product ]ö = vu, according to powers of Z)?/, if the 

 transmuted Tv of v and an enumerable infinite number of other 

 quantities, which may be called the derivatives of the transmutation 



