DOUBLE GA^r^IA FUNCTION. 
293 
(m + m') 
I((Ol) 
I(wi + Wl) 
1 
+ ve 
+ ve 
- 1 
+ ve 
- ve 
- 1 
- ve 
+ ve 
1 
- re 
- ve 
and therefore m + m = dc 1, the upper or lower signs being taken according as 
I(w^ -|- Wo) and I(<y^) have the same or opposite signs. 
§ 23 . It may now l)e shown that, if C have any arljitrary value, the function 
To I Wj, Wo) — 0 e 
CO oc 
. s X n n' 
//?! = 0 VI.2 = 0 
Lv' + 
where O = wz^w^ -f- nzow.,, will satisfy the two difference relations 
lo + Wj) _ r^(.:| Wo) - i) 
To-iGy Wo) _ Uj^lwi) 
,v.y - v: 
where m and m are the numliers previously specified, and 
n n 1 
V V' 
; 7 n 
Wi + Wo I 
-f-= log n — 
0... o 
a + 1 
_Wi w 
lU/n 
CO., 
Uog(l + “;)- 
n + 1 1 I w^ 
-log 1 + ^ 
Wi \ Wo 
+ .L „' ("i + 
J^COjCO.-) 
log W, — log Wo} 
the principal values of the logarithms being taken. 
OlDserve that with the notation previously introduced (“ Theory of the Gamma 
Function,” §§ 16 and 31 ) we may write these difference relations in the form 
Fo feg + W ,) _ FjlC I Wo) S'lfe I u,,) 
Fuk^' + COo) _ rh^iwd S',0 I .,) 
Fo“h.s) PiGi) 
The proof, to wliich we now proceed, is exactly analogous to the one just given. 
We have 
Fo + Wo) _ “1^ + V21-“l + y 22 “l Z' -f Wj j ^ J ^ ^ )*^1 ■*“ 
tf — 30 |_//Zj — 0 il/^ — 0 
+ zzpw^ + '^^OWo 
X C « + 
~zo)i H” 
20- 
where y^i and yoo are understood, as always, to mean 731(0)1, ojo) and 7.32(0)1, o)o). 
