897 



Corresponding to any positive number t, chosen arbitrarily small, 

 there is an amount S, such that 



\v^T^{u)\ <C.ny in the closed domain ((>'). 

 if ' 



\u\ <^ d , in the closed domain (q). 



From the two preceding conclusions we may infer that, correspond- 

 ing to any positive, arbitrarily small number t there is an amount 

 cf, such that 



\io^ T,l\{u)^ Tii\ <^ 't, in the closed domain («), 

 if 



\u\ «<^ (^ , in the closed domain {q). 



The transmutation T consequently is continuou.s indeed if {a) and 

 (9) are taken as corresponding numerical fields, an/i the proof of 

 the normality of' T is therefore established. 



There remains to be proved the second point of the conclusion. 



Let us consider a function u belonging to the- circle (9); as we 

 saw when proving the first point, T^ produces for such a function 

 a transmuted belonging to ((>'). 1\ {u) may moreover, according to 

 the functional theorem of MacLaurin proved in N". 15 (3"^' com- 

 munication), be represented by P^ (ti) in the domain {q'); this too 

 holds in connection with premise 3. We have therefore 





 if we indicate the coetTicients of the series l\ by P.ytOi') or simply 

 ).jc. This series converges uniformly in the domain (o') (cf. N". 4, 

 1st communication). According to the proposition of N". 18 (4^'' com- 

 munication) the transmuted v)=il\{v) of v may be found in the 

 domain («),by applying 7", term by term to the series, so that we 

 have 



roo=Vr,^A_j .(55) 







The next step we shall take consists in developing, for each 

 term of the series obtained, the transmutation in (he functional 

 series of Taylor treated in N". 19 (4^'' communication), and that 

 in such a way that we shall consider in the pi'oduct '/Mié-^', the 

 factor ï«(^) as "origin" and )J^ as "increase". According to our in- 

 vestigation in that number, and in connection with the normality 

 of 1\ mentioned before, the development in question is indeed 



