905 



circumference of (y), wliere w (.r) — x assumes its maximum modulus. 

 We can tiierefore apply the theorem inferring from it that the 

 transmutation T:=5o/, and the corresponding series P are normal, 

 with as N. F. 0. («), and as associated N. F. F. a circle which at 

 most is to be taken equal to (/?)'). It may easily be verified that 

 this is true: T = S"" is evidently a new substitution with the 

 substitution function 



tOj(^) = to|ü>(.r)| ; 

 in consequence of which the resulting series P is known together 

 with the given series P, and P,, for which reason we shall not 

 make use here of Bourlet's formula. The number that for that 

 resulting series P corresponds to a, we call ^^; /?, is then determined by 



in which .??„ is the point on the circumference of («), where to, (a-) — x 

 gets its greatest modulus. The result that the theorem produces, will 

 be in accordance with it if ^>pfi. This is indeed ihe case. In the first 

 place to (.i>.) is a point that does not lie outside the circle (y). For, 

 as already observed, the maximum modulus 



of tt), on the circumference of (a), is at most equal to y. A fortiori 

 we have therefore 



\oi{,v^)\<y, 

 and consequently to (a?„) is a point lying either within or at most 

 on the circumference óf (y). From this it follows again, in connection 

 with the meaning of xj 



\Oj{x',n) — '«'m|>| io{w{x,j)] — to(.i>.)| = |t02(.v) — H'^'3 1 • • (62) 

 • Again we infer from the signitication of x,n 



\<o{a;,n) — .v,n\>\ oi{.v,j) — ,^•„.| . ..... (63) 



The relations (60), [Ql), (62) and (63) give now rise to the 

 following reductions : 



> « -f i<^('V) — 'V I + it^sOv) ~ "»(*>! 



> a -\- \a){Xfj) — A'„.) + tOj(a;^) — to(^>)| 



which is the result required. That not only for P, but also for 

 T=S' itself, ^ may be taken as N. F. F. corresponding to (a) as 

 N. F. 0., follows again from this result, if we notice that ^^>ö^, 

 (jj being the greatest modulus of to, (x) in the domain («), and that 



^) We have comprised here point 1 and point 2 of the theorem in one expression. 



