864 
chosen, we have for the remainder Fj(7) of the series (1), after k 
terms, provided % 5 mz, 
(9) 
a 
Ry (x) eo eS) . 
d—e \B—a+d 
The amount on the righthand side of the inequality having zero 
for its limit, for £—o, the convergence of the series has been 
proved, and at the same time, as the amount in question is inde- 
pendent of x, that this convergence is uniform”). 
>. Neither in the last case should it be held that a series of 
the form (1) never produces for some function not belonging to (9) 
a transmuted in the whole domain («). Let us consider the trans- 
mutation 
wo 
7 (1—c—a)™ u de 
1 WE m —__—— 
m! 
0 
in which c is a positive constant smaller than 1. Here a, = 1—c—2 ; 
the upper limit of az on the circumference of the circle (a) with the 
origin as centre, is therefore 1 —c + u, and this is at the same 
time the npper limit for the domain (a). Hence we have 
B=1—c+ 2a. 
Choose for u the function 
U en eg | elen . {10} 
53 Lif ler 
Tu == In 
lr l—«z 
in which the series apparently converges in the whole domain (a), 
if a< 1— 3c. Further we have 8 >1, if a> tc. Finally for 
fecax<l—te, 
the corresponding 3 is greater than 1, and therefore the function (10), 
of which, in r—0, the radius of convergence is 1, does not belong 
to (3), while nevertheless the series (1) produces in the whole domain 
(a) a transmuted for that function. The reason for it is here to be 
For this we get 
1) The inequality (8) may also be used with more advantage to prove the 
unextended theorem of BourLer the series (l) not needing to be first deduced then 
from an integral. We have retained the reasoning with the integral in our cor- 
rection of the original theorem and proof in § 2, in order to remain in contact 
with the exposition of the writer, by which an easier criticism of the disputed 
points is made possible. 
