In the sequence 
Cp» Cp» Ap4t2s--- 
the numbers are therefore not ascending and as according to the 
go 
first datum the series Sw,(y) converges, we conclude from the well- 
0 
known lemma of ABEL, that 
DR oo 
ny = ern) 
p p 
is situated between (re, and Aw, where G and A’ denote succes- 
sively the upper limit and the lower one of the sums: 
wy) u(y) + Up tily)s cy u(y) is uil) ai Up+o(y) Sir axe 
If we take p large enough G and A remain below an arbitrary 
small quantity ¢, so that we have for p large enough: 
| oo 
u(e) < ea). 
Pp 
In the supposition under discussion here concerning the fourth 
oo 
datum follows out of the convergence of ~w,,(y) the uniform con- 
0 
D 
vergence of the series w,(v) for all z, satisfying a<a< y, and 
0 
as we can make y tend to 5, the uniform convergence for a <a< 6 
has been. proved. 
In the same way we might have concluded out of 
Unie) < Unb ily), 
ude) | — | u(y) 
(a = y) 
is 8) 
the uniform convergence of Zu,(e) in the range y< «<b, and 
0 
as this series converges for «=a, we should have uniform con- 
vergence in the whole interval of integration a<a<, from which 
would immediately follow what is to be proved. So we have only 
to investigate further the supposition 
| Until) | 
uw) ) | 
is) 
where the series converges uniformly in the domain a Sw < bh and 
where divergence for rv — h remains possible. 
When we put 
si 
Uy (7) 
