613 



> m h, 



m('«) 

 m! 



in which 



yl I a„ 



i-f-t 





(40) 



(89a) 



The inequality (38) holds also, according to (37) and (he tirst part of 

 (36), for all integral (not negative) k, and in all points of («), if 

 m^7u. We therefore have in {a) 



oM _ 



^"<-^^ (ti— « + t)'« , for m>m, .... (41) 



from which it follows in connection with the second part of (36) 

 that, from and after some fixed term, the terms of the series (40) 

 are in the whole domain («) comparable with those of a decreasing 

 geometrical series of positive terms not depending on ,i'. The. series 

 (40) therefore converges uniformly in («), and thus the absolute and 

 uniform convergence of the system of two successive series in (35), 

 and consequently the validity of this formula has been proved. 

 According to (39) we can write that formula as follows: 



Tiv = 2' (yif) = 7 Hi a',, 



u(»0 



(42) 



This form is the development in question of T{vii) in a "series 

 of Taylok". In oi-der to accentuate the analogy we introduce the 

 following symbolism. The development 



Tv = Vfc 





m 



suggests to represent the transmutation T by the symbol 



00 



T=yi 



k\ 



If this symbol is differentiated with respect to D as if it were a 

 power series in that letter, and if we represent the arising symbol 

 by T', we have 



T' 



= Slc^J^ 



Die 



k\ 



