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Bourtet, but Prvcaerte continually draws attention to it, though 
the notion of continuity is not expressly treated by him. The latter 
speaks of a “champ fonctionnel”; we will also use this term and 
indicate the group of functions to which the operation is to be 
applied — the domain of the independent variable (function), as we 
might call it — by the name of functional field, abbreviated to F.F. 
By way of contrast-we will sometimes call a domain of the inde- 
pendent variable number 2 a numerical field, abbreviated N.F. Two 
such fields, an N.F. and an F.F. in which the transmutation has 
been defined, we call associated. Now the F.F. will naturally always 
be of such a kind that for the functions belonging to it a common 
domain of the independent variable number a is to be indicated in 
which they are holomorphic; it will even often occur that their 
being determined in a common domain serves as defining predicate 
of the group of functions considered. But the transmutation need 
not be defined in this whole domain, as appeared already in the 
preceding pages for the complete transmutation. We have therefore 
to distinguish between the domain of z-values in which the func- 
tions are determined, and that in which the operation is defined. The 
latter we shall call: the numerical field of the operation (transmuta- 
tion); abbreviated N.F.O.; the former we call the numerical field 
of the function, abbreviated N.F.F. In most cases first the numerical 
field of operation will be fixed, in which the results of the trans- 
mutation are to be considered, and then an associated functional 
field of functions having a transmuted function in tbe first men 
tioned field. 
As appears from the paper by Frécuer it is sufficient in order to 
arrive at the notion of continuity of an operation, that in the class 
of the elements to which it is to be applied, the notion “écart” 
of two elements may have been defined; we will use the 
word distance for it. We agree that we shall take into conside- 
ration only such domains as a N.F.F. as lie entirely in finite space. 
By this agreement the cases in which all the functions of the F.F. 
have an infinite circle of convergence are not excluded from 
the considerations; the agreement only means that in such a case an 
N.F.F. must be jived that lies entirely in finite space, and to which our 
statements will refer. We further suppose that each function sepa- 
rately is /imited in the N.F.F. This contrasts with what is found in the 
memoir of Frécuet, who considers as F.F. an aggregate of functions 
which are holomorphic within the same surface, but not limited 
(Le. N°. 70); it is, however, in agreement with our preceding conside- 
rations, in which we did not in general give statements about func- 
