(529) 
of the equations (II) and that although F(2) at infinity is completely 
regular, the region of divergence G covers the whole z-plane. Finally 
there are cases in which a region of convergence Gj can be found, 
but in which the mark of the series never falls to the lowest limit 
1:k or 1:|a@|, so that there is no question about a division of G, 
into the regions G,’ and Gj’. 
Beforehand we can take care to let the series of polynomials 
possess by all means a region of convergence Gj. This takes place 
when we choose for g(u) a function always remaining finite within 
the circle | «|= 1. For then for a point a in the immediate 
vicinity of the origin « = 0 the equations (LL) 
alu A; 
will furnish for #z a value greater than 1 and the series (LIL) 
will of necessity converge in the vicinity of the origin = 0. And 
in the same way, if g(w) remains finite inside the circle |u| = k 
and the point a is lacking, the origin 2 = 0 will also be a point 
of the region G' inside which region the series converges everywhere 
with the mark 1: k. 
2. After having pointed out in general the possibility of the 
existence of the regions Go, Gi’, Gi, we must find out how to 
construct these regions. A point w belongs to G2: when one of the 
equations 3 
x g(u) = A; 
admits of a solution w of which the modulus is smaller than 1. If 
thus by the above-mentioned equation we map the w-circle with 
radius unity and centre at the origin on the z-plane, we obtain a 
region G5,; where the series diverges. It is very well possible that 
a part of the z-plane is covered bij that representation not only 
once but more times and that the representation of the circumference 
of the circle is a closed curve, cutting itself several times. Any 
region that is covered can be considered to belong to G5,5, no 
matter how. often this takes place. All the regions Go; together 
form the region of divergence G, of the series. What remains of 
the z-plane is to be regarded as the region of convergence G4, 
which we can now divide with the aid of a second conform repre- 
sentation into the two parts Gj’ and G,'. Again with the aid of 
the equation 
Zg(u) == A 
