898 



valid in the domain («), because both ?<(*) and ).k belong to (<,)'); 

 for té-^^ belongs as well as u to (o), consequently certainly to 

 (q')<[ fp), and X]c belongs to (o'), because it is a coefficient of the 

 series P^, which is complete in (o')- 



The development in question is produced by formula (42') and 

 reads M 



T[n) = 



P. (")^ + l\ (") I^K 4 — ^ D'K + 



^.(«'Ki + P\{ji)DK-^ 



2/ 



P\ (^^') 

 2! 



/>'/, + 



+ 



k\ 



+ 



.(56) 



P"M^') 



+ 



In this, every horizontal row answers to a term )-ku''^\ so that, 

 if the doubly infinite scheme is summed up according to roius, the 

 result is finite, and equal to (55j. It is necessary now to know that 

 the result is independent of the grouping, and that this holds even 

 for the triply infinite scheme that results from (56) by taking the 

 series P written at full, which we shall regularly do in future. In 

 other words we have to prove that the triple scheme is an absolutely 

 convergent one, and in this respect the proposition of the preceding 

 paragraph will be of use. 



We represent the coefficients of the series P^ by {ik, more amply 

 ju;t(^). If we replace P.jt and u^ by their natural . majorant-functions 

 ).k and nj,, and if we define by means of these latter the transmu- 

 tation l\ and /*,. which for the sake of brevity we shall call the 

 natural niajorants of Py and /\, the proposition in question states 

 that Py is complete in the domain {q), with a corresponding domain 

 smaller than (o), because this was the case with Py ; in the same 

 way f 3 is complete in («) with a corresponding domain smaller 

 than (p'). Py therefore represents just as Pi, a normal transmutation, 

 if as corresponding numerical tields, the circle ((>') (N. F. 0.) and 

 the circle (()) (N. F. F.) be taken, and 7^,, as well as P, represents 

 a normal transmutation, if the circles (a) and (o') are considered as 

 corresponding numerical fields. The transmutation Py produces there- 



1) We have written for the sake of comparison with the next scheme (56), Po and 

 its derivatives instead of Tc, and its derivatives, which, according to the functional 

 theorem of Mag Laurin, is permitted here. 



