921 



None of the four limits is, therefore, greater than [3 — a, since 

 y is at least equal to (x and at most equal to /i. Hence for the 

 whole sum (75) the limit in question is not greater either than 

 ^ — «. Thus the radius corresponding to « is for the resulting 

 series P at most equal to a -\- {^ — «)=:/?; the required result has 

 therefore been established. 



33. We may sav that with the foregoing developments our 

 original object has been perfoi-med : to ünd a symbolic formula ex- 

 pressing the coefficients of the resultant of two complete transmuting 

 series in the coefficients of these two; to fix the domain of validity 

 of this formula; finally to derive from it the statement that the 

 resultant transmutation is likewise complete ; as to the last point, we 

 found the same result with regard to the dependence between two 

 corresponding domains as was the case in the proof we gave of the 

 formula of Bourlet. 



Before, however, finishing our considerations on the subject we 

 wish to establish a few other formulae constituting with those 

 already found a sort of closed system. In the first place we have 

 in view the generalization of the formulae found in N". 31 for 

 more than two transmutations. It will appear to be sufficient if we 

 take only three transmutations, represented by the series 7^, P^ and 

 P,. We thereby assume that it is possible to assign four numbers: 

 «J, «J, «J, «J such that J\ is complete in a circular domain («/) 

 with corresponding domain («J, P^ in a domain («,) with corre- 

 sponding domain «J, P^ in a domain (or^) with corresponding 

 domain («,). Let the coefücients of the series be denoted respectively 

 by «i,7«(a')> (-i-^ini^), ^z>m{'V), those of the resultant Pjj of P, and P, 

 by ay/„„(.ï) and those of the total resultant P/jj by aiii,m{^)- 

 Then we have 



allun = {\a, I x+a, + «J'" ...... (79) 



valid in (rtj ; further, since the series P// is complete in (a,), with 

 a corresponding domain not greater than («J, we also have 



«///,m=(la//|x+rT3 + «s)'" ...... (80) 



valid in («3), and Pj// is complete in tiie domain («,), with a corre- 

 sponding domain which is at most equal to («J; all this is to 

 be inferred from (73) and what has been stated about this 

 formula. The statement that the resultant Pjn is complete so 

 that as a domain of completeness comes into account that of the 

 last component, the corresponding domain being at most equal to 

 that which for the first component corresponds to its domain of 



