926 



7/ in all points x of a domain («), centre the origin 



1 

 Urn |am|'« <a, (85) 



ni=<» 



where a^ for every integral value of m is a regular function of x 

 in that domain, then in the same points ive have for the symbolic 

 binomial expression {x -\- a)'" . 



1 

 Uyn \{x -\- a)m\^» <a -^ a (86) 



m= 00 



And as a consequence of tliis: 



If a transmutation be complete in a circidar domain («), centre 

 the origin, and the domain corresponding to («) be {S), then the 

 upper limit, 



1 

 lim I §,„!»« 



of the m^'' root of the modidus of the transmuted §„, of .f" is not 

 greater than /?. *) 



The same proposition holds if for §,„ the above mentioned majorant- 



function be substituted, the vahie of ^ being unaltered as it follows 



from the lemma in the last section of N°. 23 (5''' communication). 



If, now, we apply the preceding result to the above case, we 



find that in all points x of a domain («) not greater than («J 



1 

 /^ l§2,,«|- <« + K-«,) (87) 



since «i is the number that for the series P, corresponds to a, and 

 the difference between corresponding radii h and /? does not in- 

 crease if « diminishes. Further it may be inferred from (87) in the 

 same way as (86) followed from (85) 



1 

 lim 1(§2,,«+ ^)'«| '" < 2« 4- a, -«,. 



in-=.<x> 



This is the inequality we wished to obtain. It follows 

 that the series arising from (84') by substituting the sign -f for — 

 in (I, — x)' , and by replacing the functions §i,„ and §2,m by the above 

 mentioned majorant-functions, converges in any domain («) when 



3« -|- «1 — «2 <C "i ' or « <C i "j ' 

 because §i,,„ is regular in («J; a further explanation may be super- 

 fluous since 4t would be a repetition of what has been stated more 



Ï) For a domain of an arbitrary point Xq the same holds as to the transmuted 



of {x-—x^^)m . 



