859 
1. The condition is necessary. Suppose that it were not fulfilled 
for a certain value to, > o, so that the series 
(0) ay A et BEEREN TEGEN 
1 Sil 
ay (“,) 7 
Oo 
‘ 
diverged. Then choose a positive number v,, such that 
NS 22 
0 <0: << 01> 
and a function 
1 
= => 
vy a Ot 
5] 
whieh evidently belongs to (o). The series (1) gives for it in the 
lo, (”,) af aby iG zal) a we | ’ 
0, 0, 
point #,, 
which series, however, according to the previous observation, diverges, 
because (+) does so. In other words, if the condition mentioned ir 
the proposition is not fulfilled, the series (1) does not produce a 
transmuted in the point iv, for a// functions belonging to (v): the 
condition therefore is necessary. 
2. The condition. is sufficient. Consider a function « with a 
radius of convergence 7 >> 9. Choose a number g,, such that 
al … 
or. 
If the condition is fulfilled the series 
az) Glen) 
(yy 2) = a (4) + 
Zg (2e) 
converges absolutely for any value of z for which z—a, =0,. 
Moreover the last series, considered as a function of z, converges 
uniformly on the circle, determined by 
sei 0, dS 
for the moduli of its terms are anywhere on that circle equal to 
those of the corresponding terms, independent of z, in the absolutely 
converging series 
re a,(a,) 1 nale) ; 
0, 9; 
From this the uniformity in question may be deduced according 
to a well-known reasoning. 
The integral 
