454 



The functions a.-k have now been constructed in such a way that 

 for ail integral values of k 



On account of the additive property of T and P, in connection 

 with the fact that the quantities (pm{x) are rational integral functions 

 of A', it' follows from this that 



T{cf,,{^)) = P{<fm{^)) (27) 



Since, now, the series of functions (26) in the domain [a) con- 

 verges uniformly to u, it follows from the supposed continuity of 

 T (cf. the second form of the definition of series continuity in N°. 9, 

 2"'^ communication) that the series of functions 



T{cp,{x)), T(<p,{.c)),. .,T(r^,„ (..)),... 

 in the domain (a) converges uniformly towards Tu. The same there- 

 fore takes place according to (27) with the series 



P{<f,{x)), P{(p,{x)),...P{cf,n{^v)),..., 



which we express in the usual notation 



T{u)^ Urn P{ipA^)) , (28) 



Hi =;oo 



with the restriction that this equation is valid in the circle («). 



It appears therefore that from the premises follows that liin {P<pm {^)) 

 exists in the ivhole N. F. But the existence of P liin (y„j (x)) = P(u) ^) 

 does not follow from this. Neither had this been asserted by 

 BouRLET, who had expressly assumed that Pu exists. But even 

 if this is done it has not yet been proved that Tu = Pu, since we 

 are not allowed to conclude without more that 



lim P {fpm {x)) = P {Urn r/„. {x)) ..... (29) 



m = <x HI = 00 



This conclusion might be drawn if it were already i^nown that 

 the transmutation P is continuous in the F. F. and the N. F. 

 considered, but from the example quoted already in N°. 13, we 

 may infer that the existence of P in a pair of associated fields 

 does not imply its continuity in the same fields. Yet, some one 

 might ask : Is it perhaps possible to prove that P is continuous for 

 the special fundamental series (26), of which u is the limit, and 

 which merely consists of i^ational integral functions? In that case 

 too (29) would have been proved. 



Let us suppose therefore that the series Pu exists in the domain 

 (a) for a certain function u of F {T), of which the circle of 

 convergence (r), on account of the form of the series P, is of 

 course greater than («). We substitute, in the terms of P, for the 

 function u and its derivatives their developments in series, which 



1) Gf. the examples in Nos. 16 and 17. 



