464 



The operation of substitution furnishes simple examples of this. 

 If we have to (^) = ^ a-, a N.F.O. («) = 2(J answers to the N.B\F. (a). 

 Let us suppose è < ö< 1, then we have 1 < « < 2. The function 



uz= forms part of the F.F. of S},^, but does not belong to the 



circle («); the transmuted of it is 



^="'^*^(ï3^)=2i:^' 



(34) 



and does belong to («). 



For the series P we have here 



1 \ 1 



X ,„ ( — 1)"' 







which form becomes illusory in the point .i = l. We shall now 

 show in a direct way, that the limit (28) does exist everywhere in 

 («), and is equal to (34). 

 We have here 



(f„, = 1 + x+ x' -\- .r' + . . . . + *■'« 



- I YT = " ^ '^--^ (è)^^'^-3(è) ^^ - . . . . -ni, (I).,"" 



+ r^V4!^= (*)^^•H3(i)^^■^ + . . . . +m,(i)V« 



If these equations are added according to columns, the sum of 

 the coefficients for the {k -\- IJ^ column is equal to 



so that we arrive at 



m 

 P(.^,„(.^.))=\l(i..)A: 







This expression has indeed, since (i<^2, in rtf// points of the circle 

 («) a limit for m = oo, the latter being equal to the right-hand 

 member of (34). 



