( 305 ) 
The first property of the function Zj (a/z) is now proved at once. 
Multiplying by et we have 
I mi 
1 1 Mile arr rie 
Lim et E) mg — Fe" ze FP) Lim e «(= +1) da 
a=+o lee v a=+om 
V 
and hence, as 
eh 
Lim e Pea ye. of 
ast 
a 
4 2 Cn 2” af 
Lim ero if, (af2) = Lâm 4E = 0... (1) 
EE = n 
a=+ a=+o 0 PS +h) 
Secondly we find 
eo 1 ? de : sai le 2} a) 1 r ; 
et E, (a/z) Ten —- je ze") fe le Kak == 
. Gi. a 
0 W 0 
1 mt 
hen { oe Pe 
Ani etl 
W 
In the latter integral the only infinity of the subject of integration 
within the loop W being «= —1=e7, the loop can be contracted 
into a small circle round this point and there results 
i) 
fe (a2) da es f(z) DOR oa. Be Be 
0 
This equation may serve to evaluate f(z) for any given point z 
within the region Gj, therefore we must regard Gp, as the region 
of summability associated with the function Zp (a/z). 
Meanwhile Bors indicated still a different way to calculate 
/ (2). Supposing p = 1, and z lying within the polygon Gj, he proved 
that we have 
8n qn f 
eS Rie, 
le 2) 
Lim ee 3 
a= + & 0 Nn ! 
