310 
MR. E. IV. Bx\RNES ON THE THEORY OF THE 
Hence /x )n%"(wi + w.) — log[— (wj + "-)]} 
\XCt)| ■^*^3/ 
— [log (oJo - CO,) - log (co, - CO.)}. 
Now (^1 H~ [— ("i "o)] “ i 
according as T(coj + w.) Is positive or negative, and 
log (co.i — CO,) — ^^-^g (^1 — — d“ 
according as T(co, — co.) is positive or negative. Therefore the values of g are given 
in the followino* table :— 
I (oj, + OJ.) 
po.sitive. 
I (oJi + 0)o) 
negative. 
T(oJi — COo) 
]»ositive 
TTI. 
(0i 
TTL 
1‘- = - — 
I (dJ, — (O.i) 
negative 
TTt 
/t = _ 
(Ou 
OJ, 
In other words, 
if I Ifco,) I > I T(co.) I, g = rt TTi/co,, the upper or lower sign being taken according as 
I(co,) is positive or negative, and 
if I I(co.) I > jT(co,)i, g = ih TTt/coo, tbe upper or lower sign being taken according 
as T(co.j) is positive or negative. 
§ 33. Tbe expression for cr (z) in terms of simple and double gamma functions is 
now immediate. 
For on integrating the residt of the previous paragraph 
— log cr{z) = p pz -b log z H log r.( 2 ; | db oj,, ~ - ^ 1 ( 2 ! ±cj,) 
-Slogr,(-|±co,), 
p being the constant of integration. 
Make now z = 0, and we at once see tliat p = 0. 
Hence cr{z) = "1 
and in this expression 
— _ V v* 
nr. + Wj, +ctf.) 
■ nri-ifol ± coi)iTiVHcl ± coj ’ 
1 , - 1 (Wi + &)., 
+ log " 
ii COift). CO, ft). 
the infinities in the double summation being etpial in absolute magnitude, and that 
value of (t», w. heing taken which is on tlie same side of the real axis as CO, -f (0.1; 
while 
