658 
for every 7 the value S,'(z,). It follows thence that S,'(z,) tends to 
a limit f@(z,). 
Similarly 
Si(z)—Sn(2,) = S (zo) 
SAG) = 21 + Dz), 
(ez)? 
where lim d‚(2) =0. The first term of the right-hand side is analytical 
Jd 
SP(z) 
» 
ss : ; AG 
inside (472), its absolute value being less than Te since 
the absolute value of the coefficient of (z—z,)* in the development 
4 oo G : 
Sn(Z) = Sne) + Daz(e—z,)*, is less than BE As n increases infi- 
1 
nitely this function tends to a limit at the points 9; and therefore 
at z, too, from which it follows that a limit 
lim S",(z,) = f(z.) 
noo 
exists. 
Thus pursuing we find that for every & a limit 
lim S@(z) = f Pz). 
n= 
exists. 
3. For an arbitrary z inside (42) we have: 
Ceol? aap 
S,(2) = Sz.) zi (z—2,) S (2) ie 0000 air a Sn) (z9) ar EG (1) 
For a fixed value of z the terms of this series, if n increases 
infinitely, tend to those of the series 
(2 =) 
f (2) =d) zE (ez) f® (z,) zi 500 SF kl 
Fe.) +... (2) 
ee 
Sm 
fe) 
which represents a function, analytical in (4), since i 
Cr HA 
<a A it follows that the series (1) converges uniformly, 
if the terms are considered as functions of the two independent 
variables z and n, at the points of the set |z—z,|<3A,n=1, 2,... 
It follows from this that S, converges uniformly to / inside (4). 
Sr®(z) 
From aaa 
4. Instead of }R as well 2R could have been chosen, where 2 is 
an arbitrary positive number < 1. Hence S, converges uniformly 
to an analytical function in the interior of every circle (R) lying 
wholly inside 7. Let z be an arbitrary point in 7’, then z can be 
