(528 ) 
points x of the plane according to the values of R, belonging to it. 
In the first place we find a region G simply connected or of higher 
connectivity within which everywhere R‚ <1. That region G will 
include z == oo and the singular points A;, or one or more of these 
points lie on the boundary of this region, and in each point of G, 
there is divergence. The value of R‚ gives a notion of the rate of 
this divergence in a definite point 2, for we have 
Lim YZ | Tn(2) |= 5) - Je hoc At a CU 
By one or more closed curves Gg is separated from the region Gj 
containing the points 2, where R‚ is > 1. So this region G, is the 
region of-convergence of the series of polynomials. But not every- 
where in Gj is: the convergence equally strong. With the aid of the 
value of the limit (IV) the rate of convergence can be judged. For 
shortness’ sake we shall call that value “mark” of the series of poly- 
nomials in the point z and in general the convergence will be better 
the lower this mark falls below unity. 
On the boundary of Gy and Gj, which latter region may consist 
of different parts separated from one another, the mark is constant 
and equal to 1. If we move away from this boundary the mark 
falls. However, it never falls below 1:4, for & was the uttermost 
limit for the radius of convergence of the power series (I). Perhaps 
too, that £(oo ) is infinite or indefinite, whilst g(u) has a pole a inside 
the circle |u| =k. Then convergence is possible only for | a | >1 
and the mark of the series (III) is at least 1: |a|. 
So we shall often be able to distinguish two different parts in the 
region Gj. In the outer part Gj’ bordering everywhere on Gg the 
mark of the series is variable. It varies either between 1 and 1: &, or 
between | and 1:|a{. In the inner part Gj however, the mark 
is constant; it is continually equal to 1:k or equal to 1: |a | 
according to the nature of F'(e). The shape and size of these con- 
sidered regions is entirely dependent on the singular points of #(e) 
and of the choice of g(«). It is very well possible that Go, Gj 
or Gj are wanting. Thus for instance the series will converge 
everywhere independently of 2 with a constant mark, if in 
the finite part of the plane there are no singular points 4; 
and when [al is >1. On the other hand it may happen 
that everywhere for &, a value smaller than 1 will be found out 
1) Lim denotes here „la plus grande des limites” of Caucuy. 
