660 
For |z—z,| <4 wiz. we have 
Del : 
FES) <= (oss, eee Lee 
TE. 
0 
Here y,=|/f()—S,(| is continuous in 0O<O<27 and 2G; 
p is zero. Hence 
2 
lim [ro f(t)—S,(t)| dd = 0. 
n— co 5 
For all sufficiently large values of n therefore | /—S,,| is less than 
a given number everywhere inside (4/2), that is to say, S, converges 
uniformly to f inside (42). 
By virtue of the same lemma we have 
) k! Sp(t) ] (bHdt 
; (k) (~) — | — — fh) 
ns Sis (y= 5, a Nee fc op mile Jer = f®(2), 
(R) 
whence it follows that the series may te differentiated termwise 
infinitely often. 
Let C be a regular curve in 7’ then, again by virtue of the 
above-mentioned Jemma, we have 
Jr (ede = lim ae n(z)dz. 
Hence, if Cis ree and its anal, as well as the points enclosed, 
are all internal points of 7’, then 
[rou = 
ri 
From this also it may be concluded, according to a theorem 
enunciated by Morera’), that f is analytical throughout 7’. 
1) Reale Istituto Lombardo di Sc e lettere, Rendic., 2nd series, Vol. 19, 1886. 
