864 
ME. E. M'. BAEXES OX THE THEOEY OF THE 
§ 71. There are two alternative methods of obtaining this formula, or rather ot 
obtaining the fnndamental relation (l). 
Firstly, we may directly transform the series which expresses if;/ (z | cj, oj). 
We have, from the definition, 
— xlj.2' (2 1 w, 6 j) = 2 y.ji (oj, oj) + y.22 (w, oj) + — 
“ r 1 
■s' ■s'-' 
_1_ < V- 
+ 
0*" + (”6 + »o) « « (»*i + ?«2)“ J' 
Put now + m.. = e, a change which is equivalent to grouping together terms 
corresponding to points on the cross lines ot the figure. 
There we have 
e=l 
— i/zol (2 1 w) = yn.3 (w) + 2 yoj (oj) + -f- ^ 
= 7-22 (") + 2 : yo^ (oj) + - ^ 
e = l 
Now we have seen in § 28 that 
e -}- 1 
e -f 1 
Z -j- €0) 
ew 
~ 1 
1" 
: 8- eco 
eco 
[ 
+ 
o o 
e-o)- 
TT" 
1 ~ ) + o ■ p 
0} / &)■' 0 
, , 1 
721 (") = go 
0 ) 
722 (") = g [7 - i "]• 
CO 
We therefore have 
- ^2 (^' 1 ) " I") - g ►'^ 1 ' " I 
log w - 1 - y - . 
which is ecpiivalent to the former relation. 
Secondly, we may make use of the contour-integral exjiression in the following 
manner. 
We have (“ Theory of the Gamma Function,” § 30), when a is positive with resjject 
to CO, 
