659 
enclosed within the last of a chain of circles, all lying within 7, 
the first being (A) with centre 2, and every circle having its centre 
within the preceding one. Since the points where S, converges 
condense towards the second centre, S, converges uniformly throughout 
the second circle; similarly within all the following, hence in any 
circle with centre z which lies in 7. Every region lying with its 
boundary within 7 can be covered by a finite number of such 
circles which involves the uniform convergence of S, to an analytical 
function throughout r. 
5. Lastly we shall give a simple proof of Oscoon’s original theorem. 
According to $ 1, if S, is convergent at the points of the set (B) 
which is everywhere dense in 7’, it converges every where throughout 
T and the limiting function f is continuous in 7, whilst | /| < G. 
Now draw a circle (2) with centre z,, lying altogether in 7. If 
|z| is again <4 R, then 
Qn 
1 S,(t)dt R S,,(t) e8 dé : 
AOS aan lim il (Oe = — lip fis. t=2,-+ Refi. 
t—z OTN en (== 
(R) 
We now make use of the self-evident extension to complex 
functions of the following theorem of OsGoop *): 
If a function ~p,(), continuous in the interval a <0 <b for every 
n, converges to a function oA) which is continuous throughout this 
interval, and if, besides, | p‚(0) | < G throughout the interval and 
for every n, G being a constant, then 
b b 
lim [oO dO = | g(9) dé. 
derij: 5 
S(t) Â 2 B É 
If we put p„(0) = - , then @, is continous in 0 <6 < 2a, 
2G TO ee ae ae 
Pal < RB’ and ANS Tr ig continous in 0<@<27. Hence 
f 1 f(t) dt 
f= x [7 : 
AT t—z 
(R) 
Since f is continuous on (fe), it follows from this that / is 
analytical inside (4 2). 
The same lemma can be used to prove in a simple way that 
S, converges uniformly to / inside (4 /2). 
') W. F. Oscoop. On the non uniform convergence. Am. Journal of Math, 
1897. For an extension vide H. Lesesaue, Legons sur Integration, p. 114. 
