450 



.?;=z.r„, si elle est développable en une série ordonnée suivant les 

 puissances croissantes de x — i\, pour toute valeur de .v telle que 

 l'on air I .i' — x„ \ <[ 9." All right, it is clear now, what has to be 

 understood, when is said: a function is regular in a domain loith 

 radius q. But we do not know yet when a function is to be called 

 regular, if the emphasized words are omitted. It is not to be sup- 

 posed that by this should be meant a function that is regular in 

 the neighbourhood of any point, for in that case, a simple function 



as u = could never be object of the transmutation. Exceptional 



1 — X 



points must therefore be admitted. But how many, and how situated? 

 A function which within a certain circle is equal to 1, outside it 

 equal to 2, and on its ciicumference indefinite, is that a regular 

 function? Etc. 



All these and similar questions Bourlet has avoided by adding 

 at once, in the definition he is going to give of a regular function, 

 the words dans un domnine de rayon q. It may be assumed however, 

 that Bourlet, when he spoke of a regular function, always had in 

 view, consciously or unconsciously, a dejinite domain, in which 

 the function in question is to be regular, no matter how it behaves 

 in a domain outside it. Indeed, if matters are put in that way, 

 everything tallies. Moreover, the short parenthetic clause, thrown in 

 at random, as it were, spealis for the correctness of this view. 



We consequently assume as Bourlet's intention that the functions 

 to which the transmutation is to be applied, are regular in a certain 

 circular domain round a point .i\, no matter how such a function 

 has been defined elsewhere; further that the same holds for the 

 transmuted of these functions, and that this marks the transmutation 

 to a regular one, independent of the question whether or not it 

 produces functions with the same property for all functions that 

 are regular in a neighbourhood of -Vo. When, therefore, Bourlet 

 speaks of a regular transmutation, we shall assume that is meant: 

 regidar loitii respect to a certain functional field. 



With regard to this F. F., Bourlet tacitly makes two important 

 hypotheses. He begins his proof with the construction of the series 

 of functions 



«0' "n • • • ' «w ' (20) 



with which the series (1) is going to be built up, and he deduces 

 it from the functions into which the series of integral powers of x 



x,x^..,^x^,. («J-J 



is transformed by T. Consequently it has been tacitly supposed that 



