1113 
point « of the domain (a) the remainder fF, (7) of the series (1), 
after m: terms, satisfies the condition 
oMoe) (B—a+e\": 
Bm, (@)| < | tee 
€ J-—& B—a+d 
For the meaning of the letters we refer to No. 4. We can also 
write for this 
Rel <D X MO) 
nes eral 
d—e \P—a+d 
is an amount not depending on the chosen function w nor on 
the special point 2 of the numerical field of operation; as to the 
latter we remind of the uniformity supposition of No. 4, according 
to which corresponding to the arbitrarily given number e, an integer 
m: may be chosen, which is the same for all points a of the N.F.O. 
Let further G be an integer below which all quantities a, (z) with 
index m smaller than m-: remain in absolute value, then we have 
for each of those m-values: 
in which 
Am UI) oM(o) 
m! (9 — arti” 
and consequently 
| lj 
—~ Anu(m) 
| m ——|< EX M(), 
= m! 
in which “7 is again asnumber that does not depend on the function 
ebosen nor on a. We finally have in all points 2 of the N.F.O. 
| Tu @)| <W + EB) MQ). 
From this it follows that anumber d, corresponding independently of « 
to a given arbitrarily small number t — of which mention is made 
in the proposition of continuity of No. 11 — ean really be indi- 
cated; apparently we may write for it, 
. + T 
D+E 
The proposition has thus been proved. 
13. We observe that in the foregoing we have not proved the 
continuity of the operation in the F.F. formed by all the functions 
belonging to (8), but only this continuity with regard to the FF. of 
funetions belonging to a somewhat durger circle. But then for the 
first mentioned F.F. the proposition does not generally hold. Let us 
consider the operation for which 
