( 621 ) 
by any number of singular points 4; is to be developed into a series 
of polynomials, 
As region of convergence G, is to be considered the portion of 
the z-plane situated within the circumferences of circles with radii 
a A; 
and centers — 
a a— 
will he 
(fig. 4). The development into a series 
eh Ei 
F@)=F(0)+ > —> “mn Oe h(a—1)k , (VIII) 
ml" 
and, as in the case of series VIL the mark will always be greater 
than l:a. Only in «0 this minimum value will appear. So it 
Fig. 4 
is necessary to suppose |al 1. A division of G, into the two 
regions Gj and Gj” does not present itself. | 
The form of G for a approaching to 1 is remarkable; the circles 
enclosing G, at the same time transform themselves into right lines, 
making in the singular points 4; with the radii vectores a these 
(+5 es 
These right lines enclose a polygon, coinciding with Me. ne: 
“polygon of summatility” in the case of a—1 positive and real, where 
the angle € is a right one. At the limit such a polygon, whether 
that one of Boret or another the sides of which form an acute 
angle € with the radii vectores of the corresponding singular points 
A;, forms the region of convergence. But, if lim. a=1, for any z 
all the terms of the series VIII become infinitesimal ; then the series 
converges with the mark unity i.e. infinitely slow. 
points in the same direction of rotation an angle € = arg - 
