( 624 ) 
is limited to a closed figure, containing the origin and the boundary 
Qri 477 
of which passes through the points 4;, Aje" , Aje" ,.... In 
the supposition »=—3 and that a has a fixed, real and positive 
value that closed figure is given in fig. 5. The part common 
to all regions constructed in this way is the region of convergence 
G, of the series (LX). What becomes of Gj, if a is allowed to approach 
1 in some way or other? The closed figure in fig. 5 expands, the | 
boundary opens and, a being real, passes finally into the curve 
vu" cos n (O—a;) = rj". | 
So we see, that the greatest extension which the region of con- 
vergence G of the series (IX) with a real a can ever attain is the 
inside of the curved polygon of summability of BoreL, whose sides 
are all ares of some or other of the curves 
g” cos n (O—a;) = rj". 
By allowing complex values also for a, the region of convergence 
ean undergo here again certain modifications. In fig. 5 we can neutralize 
by this the symmetry in respect to the radius vector of 4j, and speak- 
ing generally we can cause the region of convergence G@, to expand 
more or less in a fixed direction. Such a modification will also present 
itself for a complex a if | a | approaches to 1; besides the curved 
polygon of Boren there are others of which the sides are not sym- 
metrically inclined with respect to the corresponding radii vectores of 
the singular points 4;. 
Of the development (LX) we will again give a simple example, for 
which we again take tang! z as the function to be developed. 
We obtain, if x is taken equal to 2, 
tang le 2 @ (2); 
and the function ¢p (#) is missing. 
The development (IX) here becomes 
fi ie sl F geht 
—ly,— > ies: SN SS. < 
tang—+ w# = @ + he TE as (m—1) —; (—1) hyd (a—1)4, (X) 
or 
1 3(a—1 1 #(a—1)  #(a—1)? 
ee El (a ) v(a “| JE 
a 3 a? 3 5 
ly,  a%(a—1) zal)? — a%(a—1)3 
en en TT FSI TS jk Er EE 
Eet 7 | 
Typ  a%(a—1) za — 1)? a(a—l)>  «%(a—1)4 
| ee oe as a 
at | 3 5 5 7 9 | ay 
1 a(a—l)  a®(a—1)?_— w&a—1)®_ zalt #(a—1)* 
He Ae 
aö | 3 ns B) 7 En 9 11 | + 
The development has to be compared to development (VII). The 
