( 631 ) 
We will now first of all indicate how by a modification of the 
parameters w and # the region of convergence G of the series (XI) 
can be extended indefinitely. For that purpose we put #=1 and 
use the auxiliary function 
g (u) = 1 — (1—u)¢. 
The region of the rim G," has disappeared. The region Gi’ covers 
the region G entirely and bounds immediately on Gs. The angle 
Ga in the limitation is found in e=1 (fig. 8). Within G,' the 
mark of the series 
Meo h=m 
=14+ 2 (lr E Cus! 
le jl 
ba 
AZ 
Hig?'S, 
is everywhere equal to unity and the region of convergence G’; con- 
tains two circle segments, now described on the line 01 as chord 
on either side of that line, each of them containing an angle. Se 
It is clear what happens if we allow # to approach to naught. 
The region G'} expands and if we but notice the segments becoming 
larger and larger, we shall see that finally every point « can be 
brought inside G', except those points situated on the line of + 1 
to + oe 
Passing on to the development of an arbitrary function /'() we 
come to the conclusion : ae ae @ small enough the development 
= FM (0) 
F (@) = FO = | ie = Cy A ae Sere (LE) 
i . 
or written in full 
AD 
m0 „le 0 en 1 
FO=rOtH = EIZ a arly 
(lt) 
42 
h! ml! 
Proceedings Royal Acad. Amsterdam. Vol. IV. 
