Mathematics. - "Some Consideratüms on Complete Transmutation". 

 (Seventh communication). By Dr. H. B. A. Bockwinkrl. 

 (Communicated by Prof. L. E. J. Bhouwer). 



(Goramunicaled in the meeting of April 27, 1917). 



34. As a counterpart of the formula that expresses the coefficients 

 of the resultant of a certain number of complete transmuting series 

 in terms of the coefficients of the components we shall treat of the 

 formula which expresses the quantities ^ for the resultant in terms 

 of the same quantities for the components. It is convenient 

 always to consider a neighbourhood of the origin ; or, if not, to 

 appropriate the symbol %,n to the transmuted of {x — .rj"', instead 

 of to that of x"\x^ being the common centre of the circular doniains 

 considered; if, in the latter case, we assume .r — a*, as a new variable, 

 all is reduced to the former. 



We denote the functions mentioned for the series 1\, P„ . . . 

 respectively by êi„„ §,„«,•• , the resultant for two, three,... series 

 by Hll,m, ^III,m, • • • ; fnrlher we retain the notations and suppositions 

 of the preceding paragraph. Taking fiist two components, we have 



êll.n = P, I\ (^"') = J\ (i,„n) 



The function §,,„, (.t) belongs to the circle («j), thus we iiave by 

 (67) in the domain («J 



^//>m = bn». (.^- 4- a^)== y I — — , • , . (84) 







where the last member expresses the proper meaning. Now a^i can 

 be expressed by means of the functions f,„„ according io the sym- 

 bolic formula (23), giving 



?„,... = j^^ (84.) 







provided the substitution of (^'j — xy for a^,i provisionally does not 

 mean any other thing than a de/inite mode of calculation of a^,^ 

 considered as a whole. We now notice again that a corresponding 

 formula, tuith the very same domain of validity, applies in the 

 case that arises from the present one by replacing the functions 



