426 
/ (2) is given, such that the limiting function f (2) has a finite derivative 
Jy 
at 0, and if moreover the series f,* (0, z) + f,*(O, 2) +... converges 
quasi-uniformly at 0< |z| <b <a, then it is possible to assemble the 
terms of the series in groups in such a way that the new series may 
be differentiated termwise at 0. 
If we put > f (2) =—S,(z) and = f* (0, z) = Sy (0,2), then Sy’ (0, z) 
1 n 1 n 
is continuous for OS |z| <b and SF (0,0) = S„ (0). 
Besides, for |z| > 0 we have 
lim S# (0, 2) Ne) path gd 
By hypothesis this function mere to a finite limit co as z2—> 0. 
Now let an arbitrary number ¢, > 0 be given. We construct a 
set of points z,,2,,... where lim z,— 0. From a limited number 
=n 
of indices an index can be chosen at each of these points such that 
SO, z)—f 0, 2) < 
Hence an infinite number of points z, exists where one and the 
same index can be used which we denote by ,. As z, tends to 
zero Sn, (0, z,) tends to S'n, (0) and f*(O, z,) to f (0). Thence it follows 
that | S',,(0)—/'O)| Se. Let ¢,,¢,,... BE a KERK sequence of 
positive numbers, having zero as limit. It is again possible to choose 
for every z, out of a limited number of indices an index n, >> n, 
such that | Sn, (0, z,) —f*(O, z | <€,, whence may be concluded, as 
ny, ( 
before, to the Ee of an index n, >> n, such that | S',,(0)— f'(O)| Se, 
Thus pursuing we find that there is a partial series of ee 
S” (2), S! (2),.-., such that lm S (0) = 7"), which establishes the 
p=n p 
theorem *). 
12. The theorem of the foregoing § may be reversed as follows: 
If within \2) <a a convergent sequence of analytical functions f. (2) 
is given such that the limiting function f(z) is continuous for z 4 0 
and has a finite derivative f’ (0) at 0, and if moreover a partial 
sequence Jes (2), /, „ (ress can be found where ae f,©) == f (0); then 
the sequence of functions f* (0,2) converges quasi-uniformly for 
O< \2| <b, where 6 is an arbitrary number < a. 
We denote by fe and N two given positive numbers. Then a number 
n, > NV can be determined such that 
FOL) <e. 
1) Verh. Ac. v. W., vol. 27 (1919), p. 1102. 
