423 
For wy (a, y, 2) may be written ® (#, z)—® (y, 2), where 
Tn (2) —f;, (v) (2) 
UV 
Since the functions f are for z| <a analytical w (w,y,z) is an 
analytical function of «, y and z in the region |x| <a, |y| <a, 
|z|<ca, if only for uv we put P (u,v) = ®*(u), which has been 
P (u,v) = 
tacitly assumed in the above. 
6. Let it be given that f is analytical. Let (x,, y,,2,) denote a 
pont sof “7, and’ zj z,, then 4, 4 z,. 
We then have 
Lis (Yo- 24) (#4) + (wy) (40) + (2, wy) f (29) 
wae. (2... 7 ,. 2.)== == UE sere) 
n=o ” (@— Zo) (Yo Ze) 
In a sufficiently small neighbourhood @' of (#, y,2z,) we have also 
I; _ gs) f(a) + (le) fF) + (ee) f (4) 
Ra (237, 2) = — : gs 
n=o * («—z) (y—2) 
The being analytical of f involves the same of y (#, y, 2) within 
2' and in particular the continuity of w(a, y, z). If e and N are 
given then an index MN, > N exists such that 
| Wy, (or Yor Zo) — W(X, Yor Zo) | CE 
From the continuity of wp 
(B 2): 
N and w within £' it follows that in 
0 
&' is contained a region @2 in which everywhere 
[Wy (1m 2)—P@y2z)|Ce--- 2 2. - (3) 
If n= Ye then w (,, ¥,%) =9 for every n. Also wp is 
continuous in a neighbourhood @' of (#,, ¥,2,) and moreover we 
have lim w (, 4, 2) = 0, so that here too the inequality (3) is satisfied 
dnek 
= 
Zo 
in a neighbourhood 2 of (w,, y,, 2,): 
Every point of V therefore lies in a neighbourhood 2 such that 
throughout $V by one and the same index J, the relation (3) is 
satisfied. Since V is limited and closed V can be covered by a 
finite number of these neighbourhoods’), whence it follows that yp. 
converges quasi-uniformly at the points of V. 
7. Conversely, let be given the quasi-uniform convergence of y 
at the points of every V. Let z, denote an arbitrary point where 
1) E. Bore. Legons sur les Fonctions monogénes, p. 11. 
28 
Proceedings Royal Acad. Amsterdam. Vol. XXII. 
