612 



together, we can, at least formally, represent the transmuted Tio 

 as follows 



^J'^ = 2Z2^^nr+l^^-Ji' ..... (35) 

 



We shall now assume that it and v both belong to {^), and then 

 prove that the doubly infinite series in the right-hand member con- 

 verges absolutely, then, as is known, the validity of the change in 

 the grouping of the terms, has been proved, and consequently the 

 correctness of the last formula. 



If u and V belong to 0?), they belong also to a somewhat greater 

 circle {q) with a radius q =z d -\- ó, (rf^O). If further the maximum 

 modulus of u as well as of v on the circumference of [()) be smaller 

 than M, we have in the domain (a) 



Since, further, the transmutation is complete in {a) with correspond- 

 ing douiain {(i), there con-esponds to any given arbitrarily small 

 number e, an integer iii valid ^) for the whole domain («), such that 

 I «» I <(t^ ^«^f)'S for n> n- .... (37) 



Consequently we have for k >_ rii, corresponding to any integer m 

 (not negative) and in all points of («) 





If we now assume s chosen smaller than ö, the left-hand member 

 of this inequality is in the domain («) com|:farable with the general 

 term of a decreasing geometrical series of constant positive terras. 

 The series 







(39) 



converges therefore absolutely (and uniformly) in the domain («). 

 Since the first series in (35) arises from it by multiplying the terms 

 by the factor u('"' -. mf, which is independent of k and limited in 

 (a), the latter converges as well absolutely in («) for all integral 

 (not negative) values of m. 



There still remains to be proved the convergence of the series 



^ See the so-called uniformity supposition in the 2nd paragraph of N°. 4, 1st 

 cora muni cation. 



