462 



of the corresponding series P we have èm = w", which, substituted 

 into (24) gives 



S„>{u) answers therefore to the "series of Mac-Laurin" 



* (to— .r)'« , . 

 ^^ mf 







The functions a,n{^) are regular in (a), since it had been assumed 



that a is smaller than the radius of convergence A of to(A'); the 



number 



1 



ax = lint I a,n |'" = | tu — t'C | 

 is limited in («) and its upper limit is 



a («) =r i a> {Xm) -^ ^,n i , 



if A'„i is the point on the circumference of («), where «^r attains its 

 maximum value. The series Pu represents therefore a transmutation 

 that is complete in (a), with a corresponding domain (^) the radius 

 of which is determined by 



^ = « + 1 CO {Xm) Xm\- 



The F. F. of P is in most cases a part of the one of *Sw, because 

 as a rule ^^o. For we have in gene^^al, if x^ is the point on the 

 circumference of («) where to (.r) assumes its maximum modulus tj, 



^= \Xm 1 + |«> ('■»m) — ^m I > l^^l + | «> (-ï^^) ^tI 



^ j iCj -j- CO (^ j) — x,\ = \(C {x^) i =r tJ. 



If, however, to(^,„) has the same argument as Xm, ti is equal to 

 |to(^TO)l; the result /3 ^ <J or 



|cü {x,n)\ > jw (.■?;^)| 

 would then be inconsistent with the meaning of a> (.r^) ; in this case 

 the points x,n and x^ must coincide, as well as the circles {a) and (/?j, 

 so that now the F. F. of P is exactly equal to the one of S^>. 



In any case all functions belonging to (^) form part of the F. F. 

 of S; therefore according to the theorem proved we have for those 

 functions, in the domain («), 



S,,{u) = P{u). 



We already observed in the discussion of an example of this 

 series P, in N°. 7 (2"^ communication) that it might be deduced at 

 once from the ordinary theorem of Taylor of the theory of functions. 

 Indeed, if we write 



S.J, {u) r= u [co (x)] -^= u [.?; + (tt> {x) — x)] , 

 it appears that S'.,u in the point x^ of the domain («) may be 

 developed in the series of Taylor 



