560 
taken along the circumference of (0,), exists therefore, and may be 
found by term by term integration. The series (1) is the result 
then, and this series converges therefore, for the function considered, 
in the point «,. The condition is thus sufficient *). 
3. With regard to the proposition we observe the following. 
There will evidently be a lower boundary a for the numbers y, 
such that sonvergence of (3) takes place for any ¢ with modulus 
greater than a, and divergence for any ¢ with modulus less than a; 
this again follows from the shape of a power series, which (3) has 
1 ane 
with regard to —. At the same time it is to be deduced in the 
t 
usual manner that the numbers a,,(v,) satisfy the following two 
conditions : 
1. Corresponding to any arbitrarily small number « there is an 
integer m-:, such that 
am («,)| < (a + 8)", for m5 me. 
2. Corresponding to any arbitrarily small number « there are 
an infinite number of integers m, for which 
Gm (x,)| > (a --€)". 
On the other hand these two conditions are necessary and sufficient 
to characterize the number @ as the lower boundary mentioned above. 
According to a well-known mode of expression we can also say 
] 
that a is the upper limit, for m= om, of the expression a (7,) m 
i.e. symbolically 
E sone 
a= tinae jj EER ME 
m—P 
s 
JOURLET’s theorem may now be expressed as follows: 
If the series (1) is to produce in a given point x, a transmuted 
for all the functions belonging to a certain domain (9) with centre 
it is necessary and sufficient that 9 is not smaller than the 
hl 
7 
Re 
number a determined by (5). 
The circle (a) is therefore the minimum circle with centre x,, 
of which it holds that the series (A) produces a transmuted in «, for 
all functions belonging to it; this statement is again equivalent ta 
the theorem of Bourrer. It ought however not to be supposed, that 
a series of the form (1) never produces a transmuted in 2, for some 
1) See the observation in note 1 of § 4 about this proof. 
*) Ene. d. Math. Wiss. I A. 3, p. 71 
