914 



7"("')(y)7<("<;(.r) 

 in! 

 But this is, if we consider v {x) as "original point" and u{.v) as 

 "increment", exactly the general term of the series in question, the 

 validity of which we proved in the 4^'' communication. This proof), 

 as a matter of course, consists in shewing that the change in the term- 

 grouping is permitted, the conveigence of the aggregate being 

 absolute. It may therefore be grouped in an arbitrary manner so 

 that the symbolic formula (69) admits of the following interpretation : 

 replace both functions v {x -\- a) and u {x -\- a) by their power-series 

 in the letter n, then form the aggregate arising from the multiplicative 

 combination of the series-terms, and substitute indices for exponents. 

 If the so obtained aggregate be ordered according to indices of a 

 we simply get the functional series of Mac-Laurin for Tw = T{vu); 

 if it be ordered according to powers of Du, the functional series of 

 Taylor for Tw in a "neighbourhood" of iv = v is obtained; if, 

 lastly, the aggregate in question should be ordered according to 

 powers of Dv, we should (ind the functional series of Taylor for 

 a "jieigbourhood" of iv =i u. The symbolic formula (69) contains 

 all these different cases; we only wish to observe that, if we expand 

 the right-hand member according to powers of Du, the general 

 coefficient in that expansion, which is, except for the factor 1/m!, 

 equal to 



a"' v{x -j- o), 

 or to J'W {v), has in this very form a meaning only in domains («) 

 smaller than (r,), where r^ is the «-value to which the radius of 

 convergence r of the function v corresponds as a |?-value; whereas 

 in N°. 20 we saw that the other form of the coefficients in question, 

 viz. that defined by (45), possibly has a meaning in domains greater 

 than (rj. 



31. We now come to our principal object; to construct a sym- 

 bolic formula which expresses the coefficients dm of the series P 

 answering to the composed transmutation 7'= T^T^ in terms of 

 the coefficients ).m and ƒ!,„ of the partial series Pi and P,. As we 

 said already, first the functions 



U^T,T^{x% 

 into which T transforms the integral powers of x, are determined 

 for the purpose, in order to derive from them, by means of 



1) We wish to insert here the remark that the proof we refer to becomes simpler 

 if the majorant-functions am of am are used, as we did in Itie 5tli communication. 



