3 
which are the transmutated of the successive positive integral powers 
of x: the latter functions exist according to 2 and belong to the 
circle («). The above-mentioned theorem of Mac Lavrin consists in 
stating the equality of the transmutations 7’ and P, in a certain 
numerical field (a), which is part of (@) or identical with (©), 
if the further condition is added, that the latter transmutation is 
complete in (a’)’). 
In this statement there is something unsatisfying, if we compare 
it with Tavror’s theorem for the Zheory of Functions. The latter 
asserts: “if a function, in a certain circle, has some specified proper- 
ties (viz. a definite differential coefficient) it can be expanded in 
TayLor’s series in that circle”. It is therefore not necessary to impose 
further conditions on that series. Accordingly it would be desirable 
that also in the Functional Calculus the theorem might be stated in 
such a way that it were not necessary to impose any further con- 
dition on the series P corresponding to the given transmutation 7’, 
but that such conditions were implied in the properties of T. At the 
time it was our opinion that this was not the case. But now we 
are in a position to prove the following proposition: 
The series corresponding to a normal additive transmutation 
represents a complete transmutation. | 
For simplicity we consider a circular domain (7) round the origin 
as a centre and in this the infinite sequence of functions 
IEEE ta ETE Senate (4) 
to which, by definition, the infinite sequence of transmuted 
en L.A 
corresponds, the latter functions being regular in a circular domain 
(a). If we denote by e an arbitrarily small positive quantity, then 
the sequence of functions 
a x? ym 
Pir (Gree ool 
derived from (4) converges uniformly to zero in the domain (a). 
According to a simple property of continuity (le. IJ, n°. 11) the 
sequence of the transmuted of the latter functions, which, by the 
additive property of the transmutation, is represented by 
(6) 
1) A transmutation P, represented by a series of the form (1), is called by me 
complete in a domain (a), if there is a certain circle (¢), concentric with (4), 
such that all functions belonging to (o) possess a transmuted, regular in (~). The 
minimum circle (2), which may be taken for (¢), 1 called the domain corresponding 
to (a) (lc. Vol. XIX, N°. 6). 
1* 
