Fig. 1, 
If R, is the modulus of u deduced from the above-mentioned equa- 
tion, then 1: R, is the mark of the series in the point z. We find 
then the mark is constant on circumferences of circles concentric 
with Gj and it can vary on each radius between 0 and 1. 
Just as this will present itself in more general cases, the auxiliar 
function g(u) contains here a parameter a, which can be varied 
arbitrarily either for a definite point « to lower the mark of the 
series i. e. to strengthen its convergence, either to enlarge or to 
reduce the region of convergence Gj. In the considered very simple 
case all this is easy to see. Considering in the first place the trans- 
formation and also the situation of the region of convergence, we 
see that for Gja==o transforms itself into the unity circle and that 
the series (VI) again becomes the ordinary power series. 
When |a—l | decreases the circle G, becomes larger and larger, 
at last when a approaches to unity G encloses half of the z-plane, it 
1 
has become the half plane containing the point — zi (M) and limited 
a— 
by the normal, which can be erected in the point e =1(A) to 
AM (fig. 1). But this is a limiting case, for then in G, the mark of 
the series is everywhere 1; independent of x the terms of the series 
