457 



We therefore suppose, more exactly, that the transmutation P is 

 complete in the circular domain («), with centre n\, which forms 

 the numerical tield of operation of T; further that there are among 

 the functions belonging to the domain (/?) corresponding to («), such 

 as are contained in the F.F. F{T) of T; the aggregate of these 

 functions we represent by i^,(T). The equation (28) now holds in 

 the N. F. 0. («) of T for all functions contained in the F. F. of T. 

 A fortiori the equation holds in the domain («) for a function u of 

 F,{T). But we further have 



Urn P{rf,n{x)) = Pu ....... (29a) 



m=co 



For 1° there is, because u belongs to (j3), a circle (p) > (|3), in 

 which the fundamental series (26) converges uniformly towards u,. 

 and 2° the series P, on account of its completeness in («) with 

 regard to {^), is continuous in any associated pair of fields, of which 

 («) is the N.F. and the F.F. consists of the functions belonging to 

 {qj (cf. N". 12, 2"^^ communication). 



From (28) and (29) it now follows that 



Tiu) = P{u) • 

 this equation being valid in the domain («) and for the functions of 



We now have proved a limited validity of Mac-Laurin's ^) theorem 

 for the functional calculus, which is expressed in the following 

 proposition : 



Jf the series P, ansioering to a normal additive transmutation T, 

 is complete in the circular domain («), lühick foi'ms the N.F.O. 

 of T, loe have in this domain 



T{u) = P{a) 



for such functions of the functional field F{T) of T as belong to 

 the circle {{f) corresponding to (a). 



With this for the first time, in so far as I know, an exactly 

 defined category of cases has been given, in which Taylor's theorem 

 for the functional calculus may be stated. 



It follows from it that, before we can proceed to the treatment 

 of this theorem, we have first to consider those series of the form 

 (1), which represent a complete transmutation. The treatment of 

 theorem XI of Bourlet must therefore precede the one of theorem X. 



Observations 1^^ We have not violated the generality by suppos- 



') We call the proposition especially theorem of Mac-Laurin when we oppose 

 it as a special case to the general theorem, of Taylok, which we are going to 

 treat in N^ 19. 



30 



Proceedings Royal Acad. Amsterdam. Vol. XX. 



