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that minimum distance from «,, the circular domain of S,,,) will be 
exactly the same as that of the corresponding series, at least in so 
far as regards functions with no other singular points than s. 
If we consider a group of functions belonging to the circle (1) 
with «, as centre, each point of this circle must in turn be 
considered as a singular point, and, if 2, always lies in the domain in 
which a, <|}s—a|, the minimum distance 7’ from z, to the aggregate 
of all the circumferences as mentioned above has to be determined ; 
(r’) is then the circular domain of operation of the series corresponding 
to 7’ for the functions in question, while the circular domain of opera- 
tion round z, for 7’ itself is in general greater, having a radius equal 
to the minimum distance of z, to the (possibly moved) singular points. 
8. In the considerations of the preceding number it was supposed 
that the transmutation 7’ had been defined in another way than by 
the series (1). It may, however, occur that we start from a series 
(1) as the definition of a transmutation. The latter then gives for the 
moment, for functions with a circle of convergence (9), a transmuted 
function only in the interior of the corresponding domain («). But 
the analytical continuation of the function v= Tu, now furnishes so 
to say at the same time the analytical continuation of the 
transmutation, so that the latter is also determined outside the 
domain («). The continuation thus considered however has to be 
repeated for each new function, while it would be desirable to 
have an analytical expression, which represents the results of the 
operation, at least for a complete sub-group of the functions 
considered, in a domain outside (@). 
In order to carry this out, a point wv, inside the circle with 
centre z, may be taken as a new centre and to that purpose a 
correspondence between quantities «@ and @' as was explained in 
N°. 6, may be established; the functions a, («), from the given 
series supply the means for it. Not for all the functions, however, 
belonging to (2) the series will converge in the new ‚domain («), 
so that this proceeding, in anticipation of an expression that we 
shall introduce in the next number, will always furnish an extension 
of the numerical field at the cost of the functional one. We observed 
the same in considering concentric fields: if a increases 3 does, i.e. 
the functional group shrinks. 
The consideration in the preceding number teaches, however, 
that in this way, even if only functions with one and the same 
singular point are taken in view, a whole field of points cannot 
be reached where the transmuted of those functions really exist. 
