858 
pour toute valeur de « telle que lon ait (wr) << 0,” (with only 
the symbol <, not the symbol = with it). Besides, if Bovrrer, in 
stating his theorem XI, had meant functions that are also regular 
on the circumference of (v), the first part of his proof would have 
been wrong. For in that case the function ‘B) need not have a 
transmuted in the point .7,. 
If we stick to the term ‘regular’ as laid down by Bouruer himself 
and in frequent use, the second part of his proof is incorrect. But 
this ought not to be wondered at, if we observe that the corre- 
sponding part of the theorem is also false. 
Let us consider the transmutation 
Tu === m TEE ee he 5 . . ‘ . . (2) 
The series (A) is here, for a domain of the origin (7, = 0), 
. l 1 1 
WO 1 at | 
= ie EROP 
BE (m-+-1)?2™ 
and eonverges for a// values of z with modulus 1. For the function 
1 
UZ, 
(1-—a)? 
that is regular within the circle with radius 1, the series (2) produces 
however no transmuted in the origin. . 
2. Although the inaccuracy in the stating of the theorem is slight, 
it seems proper to express it in the following more accurate form : 
If the series (1) is to produce for all functions belonging to a 
certain circle (0) a transmuted in the centre «x, of that circle, it ús 
0 
necessary and sufficient that the series 
j ' «,(%,) a,(@,) | 
Westie hd DE —— air {= dy a> a ee 
converges for each value of t with modulus greater than 9. 
Before proceeding to the demonstration it is convenient to make the 
following observation: From the shape of a power series which (3) has 
with regard to — , it is to be deduced that, if it converges for a certain 
t 
value of f, it converges absolutely for any other value with greater 
modulus and, if it diverges for a certain value of ¢, it diverges for 
any other value with smaller modulus. 
The necessity and sufficiency of the condition is now proved in 
this way : | 
