( S56 ) 
By taking p large enough we can make |G’ and A’ according 
to the second datum smaller than an arbitrary small quantity e and 
for p large enough we have thus at the same time 
b 
Lee | 
> furn(y)du) Ze, 
| 
|p 
t 
oo | 
ES fulx)de < Spee. 
Pp 
a 
By these inequalities is expressed that the series 
t 
oo 
= | u,(2)da 
0 
a 
converges with respect to ¢ uniformly in the domain a<t<6. We 
es} 
have already proved that in the domain a<a< b the series © u(x) 
0 
converges uniformly, so that the equation 
t t 
oo Ld 
í F(a)da = > { useyde 
3 0 
a a 
certainly holds, and when we then finally apply the principal property 
of the uniformly convergent series we find when ¢ tends to 6 
b t b 
. oo DD 
fra = Lim = | u,(z)de = = Fur(e)de. 
- i=6. “0 0 
a a a 
With this we have given the proof of the enunciated theorem and 
it is clear that the proof holds if the third and fourth data only 
hold for all numbers 7 surpassing a definite number. 
For the evaluation of the integral 
NEL 
Ea 1 (u =) 
Mys! x 
É =f) dx 
Jel 142 
0 0 
(where s >> 0) the theorem can be applied. 
We have 
Ns! 
(u -) 1 mies) 
—( ty =) = 
1l-+@ KN 
HMMs 
1 Ns! 
(—1)" (ty =) we, 
ie 
and the series converges for O<w <1. The terms do not change 
