1107 
This domain is, generally speaking, the same for all functions with 
the same singular points; exceptions apply to cases for which 
Bourrer’s theorem, does not hold as was indicated at the end of 
N°. 3. If, however, the result v= Zw has been continued analytically 
for one function w of the group, and if the same has been done 
with the results of certain operations that may be called the derived 
operations of 7, it is possible to indicate for all the functions 
having the same kind of singularity as u, a series containing the 
results mentioned as coefficients, and converging in parts of the 
excluded domain. If conceived in an opposite direction, this process 
produces an eatension of the functional field with conservation of 
the numerical one; it forms the proper analogon for the functional 
calculus of the “prolongement analytique” in the ordinary theory 
of funetions. (See further N°. 20). 
We observe, that a regular transmutation may only be continued 
in one way, as this is the case with a regular function. 
9. We will now discuss the question in how far the complete 
transmutation is continuous. 
Bourtet has called a transmutation 7’ continuous “si la limite 
de la transmuée d'une fonction est la transmuée de la limite de 
cette fonction” He explains this further by adding: “En d’autres termes 
si une fonction w(x, h), dépendant d'un parametre A, tend vers une 
certaine limite, lorsque A tend vers une certaine valeur, Tu (7, h) a 
aussi une limite, et lon a 
lim {Tu(#,h)) = T [lm a (@,h)\-. 
In this it is implied that all the functions to be considered may 
be obtained by attributing a definite value to the parameter / in a 
certain expression u (w, h), this value belonging to a certain compiex 
domain, in which 4 varies. It is however useful in connection with 
the character of the complete transmutation to make a somewhat 
more general supposition about the group of functions to which the 
operation is to be applied, more in accordance with (though not 
exactly equal to) the one to be found in the paper by M. Frécner 
“Sur quelques points du calcul fonetionnel” (Rendic. d. Cire. Mat d. 
Palermo, 22 (1906) p. 45 (N°. 70). 
Before, however, proceeding to a more precise statement of the 
nature of the group of functions in question we will make a few 
general observations, in which we have not specially in view the 
complete transmutation, but an arbitrary additive operation. 
That it is necessary to indicate very exactly the group of functions 
over which the operation is to be extended, is not mentioned by 
