459 



there is a number a, such that in the whole field («) 



|Z)-i(?0| <r, if Mu<ö, 

 where Mu is the maximum modulus of u in that same domain («), 

 which here represents both the N. F. F. and the N. F. 0. of T. 

 D~^ is a fortiori continuous with regard to the same N. F. («) in 

 any F. F. of functions belonging to a greater circle than («). 

 For the construction of the series P we have 









If this result is substituted in the formula (24), we tind, if m^ 

 indicates the binomial coefficient of m for the ordinal number k, 



_^ m-k __ ^A+l 



«;« = >^ 'rr^khi — ^^) = 2^ ^^ TTTT ^ ~ ^)"'~^" 



~'~ 



= —^^ Vfc (m4-l)i+i(- 1)^+1. 



* 



We now have 



m m-\-l 



yi (m+l)fc+i(— 1)^+1= V (rn+l)/(— 1V= (1— l)'"+i =r 0. 



— 1 



so that 



m 

 



Consequently 



( — l)'»x'>^+^ 



am= — . 



m-\-\ 



D-^ II therefore answers to the "series of Mac-Laurin" 

 The upper limit 



m-4~l m! 



' 



1 



«a- z=z Urn \am \^ (6) 



m = 00 



is here equal to |.r| ; the upper limit a (a) of a^-, for the domain («), 

 is therefore equal to a, and the transmutation is complete in the 

 domain («), with a corresponding domain (^), the radius of which, 

 according to the formula (7) (1^' communication), is determined by 



^ ^ a -\~ « = 2a. 



30* 



