Mathematics. — "Some Considerations on Complete Transmutation''. 

 (Sixth Communication). By Dr. H 1^ A. Bockwinkel. (Com- 

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



(Communicated in the meeting of Marcli 31, 1917). 



In the preceding communication we treated of the transmutation 

 T= T^ 7\, which is obtained when two complete transmutations 

 7\ and 7', are a[)piied lo some regular function u. We saw there 

 that the resulting transmutation is likewise complete in some pair or 

 other of associated fields, the mutual dependence of the new N. F. 0. 

 and the new N. F. F. being to some extent established. We further 

 gave a strong pi-oof of the formula determining the resulting series 

 P, which was furnished by Bouri,et without domains of validity 

 being mentioned by this author. As we have seen the formula 

 expresses the so-called operative function of the resulting series P 

 in those of the components P, and l\ and ditferential coefllicients of 

 them. Again in giving some examples to illustrate our theorem of 

 N". 24, we observed that the method to find the resulting series by 

 means of the just mentioned formula of Bourlet, is often much 

 more difficult in practice than a somewhat more direct method, 

 according to which tirst the functions §„, (x) =z T, 1\ (.?;'") are determined, 

 and then, by the symbolic formula (24) 



a,„ == (5 — .r)'« (24) 



the coefïicients a,n {x) of the resulting series P. Bourlet, however, 

 has been able to apply his formula with success to questions of a 

 more theoretical character. 



The examples mentioned give rise to the question whether it is 

 possible by means of the more direct method to find a general 

 formula which expresses the coefficients üm of the resulting series 

 P in the coefficients ;.,„ and fi„, of tlie composing series. We thus 

 arrived at a rather simple symbolic formula, which allowed us to 

 shew again the completeness of P, the statement about correspond- 

 ing domains being the same as in the foregoing communication. 

 The investigations which led us to this result, gave us an opportunity 

 to establish other more simple formulae, which served us to go on 

 further, and which have moreover a certain interest in themselves. 

 Again it seemed convenient to add some further formulae to those 



