ed 5, 5, pri ate 6 
$0) ’ OSC Roe ON . « . . ( ) 
o+eé (6 +) (6 + er 
will converge uniformly to zero in the domain (a); because a normal 
transmutation is continuous in a pair of conjugate fields. For suf- 
ficiently large m-values we have therefore in all points of (a) 
El lo sy ELT ON 
From the equation (2) it is now easily derived that an analogous 
inequality holds for the functions a, occurring in the series (1), that 
is to say, that these functions, too, are less in value than the mt! 
power of a certain number independent of m. For if (7) is valid 
for m > m,, we have by (2), |2| being at most equal to a, 
mo m 
|am| < Sant Exel + Se mpam—* (6 +. e}k 
0 mo +1 
m 
GI “le Ni a 
0 
<i + (5 Hate 
The first part of the second member of this inequality consists of 
a fixed number of terms, each of which is, for sufficiently large 
m-values, less than (a + €)”, so that the same holds for their sum. 
The second part is greater than the latter quantity, hence we have 
for sufficiently large m-values at all points of the domain («) 
ln (Orb ED vern eis oa ee 
and therefore also 
de 
lim lan) Gore eeen GRRER 
Thus the upper limit in the left-hand member of this inequality 
is finite, and this is (Vol. XIX, N°. 6) the very condition under which 
the transmuting series (1) is complete in the domain («); moreover 
we infer that the corresponding domain (3) has a radius 2, no greater 
than o + 2a. For all functions w belonging to the circle (o4-2e) the 
series P therefore produces a transmuted function Pu in the domain 
(a), and this transmuted is equal to Zu, according to the functional 
theorem of Mac Laurin we gave in the form (Vol. XX, N°. 3): 
Lf the series P, answering to a normal additive transmutation T, 
is complete in the circular domain (a) forming the N. F. O. of T, 
we have in this domain Tu = Pu for such functions of the functional 
field F(T) of T as belong to the circle (8) corresponding to (a). 
