Mathematics. — "Some Considerations 07i Complete Transmutation." 

 (Fourth Communication). By Dr. H. B. A. Bockwinkel. (Com- 

 municated by Prof. L. E. J. Bkouwer). 



(Gommunicaled in the meeting of December 21, J 916.) 



18. In this paper we sliall discuss tlie general theorem of Taylor 

 for the functional cak'ulus, a particular case of which, that we, as 

 such, called AIac Laurin's theorem, was discussed in the preceding 

 paper. Befoie proceeding to ihis, we shall, however, treat an impor- 

 tant proposition of the normal additive transmutation of which we 

 shall avail oui-selves in the future. 



The proposition in question is a special case of another that holds 

 in general foi- a coutiriuous traiismutatiou and which, generally 

 expressed, states, that such a transmutation /tde^óMts additive property 

 with regard to an injinite sum. The proposition is as follows 



If a S'ivies the terms of lohich are fuïictions that form part of 

 the F.F. of a cont/.n/tums nddiilve transmutation T, converges 

 uniformly in the JST.F.F. totoards a function u, lohicli also 

 forms part of the F.F., the ■^eiies the terms of ivhich are equal to 

 the transmuted of the first mentioned terms, converges uniformly 

 in the N.F.O. toivards T{u). 



For if we have 



u =r w„ 4- Wj + . . + ?<„i + . . ., 



such that the series converges uniformly in the N.F.F. of 1\ this 

 means: corresponding to any given, arbitrarily small amount cT there 

 is an integer N such that in the ivhole N.F.F. of T 



& I 



m u,n I <^ (f, for n> N 



13 



Further, on account of the continuity of T, there corresponds 

 to any arbitrarily small amount t an amount ff such that, if v be 

 a for the rest arbitrary function of the F.F. 



I T(u) I <T in the N.F.O. of T, 



if 



\v\<^<i in the N.F.F. of T. 



Thus there corresponds to any arbitrarily small amount r an 

 integer iY, such that we have in the N.F.O. of T, 



