630 ) 
¢ 
For the points « inside this region Gj’ the equation 
2g (u) = 1 
: 1 
can be satisfied only by values of u with modulus greater than — 
u 
i 
and as g(u) becomes multiform for | ~ | < — the series of polyno- 
lu 
mials (XI) will have the mark in each point of Gj’. 
By the following consideration we can form an idea of the exten- 
sion of this region Gj’ for different values of /} and w. On the line 
oN as chord we can describe in the z-plane on either side two 
: ane 1 
circle segments, each of which containing the angle 5 The common 
ee 
radius of the segments is AN: 2 sin 5 and each point # in one of 
the segments has the property 
a—N 
= arg. 
zt &. 
As the argument of (l—u)* for every value of u within the 
ot bx 
eirele with radius — can vary but between — = 
ri 
and + om the relation 
N 
1— — = (l—w u}? 
x 
or 
2g(u)=1 
: : re 1 
is not satisfied by a single point « within the segments for | « | >—. 
u 
From this ensues that these two segments will always form part 
of the region Gj’ and this allows us to easily oversee the extension 
of the region Gj. 
If we wish to deduce from (XI) the development of a given function 
F(«) with the singular points 4; =7;e” we must then successively 
apply the transformations (#,4;) to the region of convergence 
G, = Gi + G," just found (i. e. we must let those regions rotate 
round «= 0 through an angle a; and we must let the radii vectores 
increase in a ratio of 7;: 1). 
That which is common to the transformations G,,;" of Gj or to 
the transformations G,;" of @," forms for F(x) the region Gy’ or Gj. 
As g(v) has no points of infinity within the circle with a radius 
1:4, we need not take heed of a singular point at infinity if this 
makes its appearance in F'(#). 
