452 



with oor conception of tiie nature of a functional field, as we 

 set forth in N°. 9: we stated there tliat the F. F. will of course be 

 of such a nature that all its functions are ree;ular in a common 

 domain, which we have called the numerical field of the functions 

 (N. F. F.); this is here the circle {a). It further follows from (24) 

 that all a's belong to that same circle. On the other hand, it can 

 be inferred from (23) that, if all a's belong to a circle {a), this is 

 also the case with the §'s, but Pincherle proves that the domain 

 of validity of the series (1) extends moreover over an infinite aggre- 

 gate of other, transcendent functions, of which all transmuted belong 

 to the circle in question. 



The functions am having thus been constructed, Bcurlet says: 

 "Considérons alors la série (1) 



T)( \ I ^1 , I "s (, I 



1./ 2.^ 



pour toute fonction reguliere u qui rend cette série convergente, 

 cette égalité définit une transmutation additive, uniforme, continue ^) 

 et reguliere". These statements are not further explained by him. 



Here, in the first place, the mode of expression does not seem 

 exact to me; we can say: "pour toute fonction règvAxè^YQ deXie ègdMié, 

 définit une transmutation uniforme et reguliere, but no\ : "\)0\iiY toute 

 fonction reguliere cette égalité définit une transmutation additive" or 

 "continue" ; for an operation cannot be additive for one function, 

 nor can it be continuous since for the idea "additive" at least tioo, 

 for the idea continuous, in any case an infinite number of functions 

 are wanted. The meaning, however, cannot be other than the one 

 which would be expressed if the words "pour toute fonction reguliere" 

 were replaced by "pour I'ensemble des fonctions régulières considéré." 

 But in that case it follows at once from the investigation in N". 13 

 (2"*^ communication) that the statement, at least so far as regards 

 continuity, cannot be true in its generality ; for, we found there 

 that the series Pii is convergent for all functions belonging to a circle 

 {■^f) in all the points of the corresponding domain («), but that it 

 does not represent an operation that is continuous in the functional 

 field of these functions with regard to the N.F.O., the continuity 

 being disturbed in the point x =z a. 



Since, however, the further reasoning, from w^hich it must appear 

 that Pu is identical with Tu, is based on the pretended continuity 

 of Pu, the proof must be considered as incorrect. It might be still 



') The italics are my own. The statement thai the transmutation is regular 

 cannot be objected to if the convergence of the series is uniform in the N.F. 



