DOUBLE GAMMA FUNCTION. 
297 
(2)' When 
does 
U"i+ ojo) does 
— w,, in which case 
yg.j (w^, = D d- 2 
lie within the region l)ounded by the axes to — 1 and 
{n + l)7ri 
But the cases (1) and (2)' are precisely the same, as an inspection of the figure 
shows at once, except that I(wo) is positive when I(wi) is negative, and vice versd. 
And, similarly, the cases (2) and (1)' are the same, with a similar change. 
Hence the values which must be assigned to 7oo(w^, Wo), in order that the ecpia- 
tions 
^2 + ^l) _ (1 - i ) 
(^) 27 r/cL»o 
r.-i {Z + CO,) _ I\ (d«l) (A - A 
iVH~d “v/ivk" 
may co-exist, are precisely the same. 
We shall have then these two e(|uations, provided 
733 (*^n *^3) — D, 
where D stands for 
? V' i_/,, r 1 \ "l + y j 
- a + 0 
u 0 
log [n (wj + Wo)] 
+ 
(2/!/ l)c()o —fUj 
log nwo + 
(27i -f- 1) Wj — Wo 
jQ)nCO 
otuj 
log no)^ 
(the principal values of the logarithms being taken), except in two cases, 
(1) When 1 H , ’i 1 I lie within the region bounded by the axes to 
^ U"i + "3) does not] ^ 
and — Wo, in which case 
Wo) = Li 
p) _p 2 ’ihOLuLllZid”' 
Wo 
(2) When I, ^ , !> lie within the re gion bounded by the axes to — 1 
^ ' [(w^ + Wo) does not.' ^ 
and — Wj, in which case 
733("n "2) = Li 
D + 2 
in'in + 1)' 
7Tl 
VOL. CXCVI.—A. 
2 c,! 
