MR. E. W. IR'VENES OX THE THEORY OF THE 
u 
«= 1 
{s, a I 6)|, J 
1 
— V - 
(j d -T 
a + {'»' + l)fWi + T !)&>., w, + (yo] , / .. 
^ ■' log-n(oji + a;,) 
COj^CO.-) COj^Ct).-> J 
I I T (/*' + l)&).i a)i + ct).i 1 , fft + + 1)Wi Wi + (y.i] , 
+ <1 — ■' — -" [-logncy^ -j- \ --j.loo’ w 
J [ ^1^-2 SCO^CO:, ' 
= - + :£ S' 
i) 
a 
O O 0 3 
-|- Ciy.i]^ -j- yo.-> “1~ '2 [in -j- m ^ttl , 
2a — Wj — &)o 
2a)yCo.2 
= — I coj^, 0J.2) + oSQ'(a) 2 (?}i. + rn'^TTL, 
where o ~ rn^oj-^ + 
the complete form of the result estahlished in ^ 55 for the case where a is positive 
with respect to the o’s. 
We may establish the other results of that paragraph in a similar manner. 
§ 60. Finally we will hrietly consider the reduction of the secoiid form of the double 
^ function to the doidde gamma function in the case when .s = 0. 
If we put s = e, where |e| is very small, we olRaiii 
I Oj, 0J,i) - [ct [ 0|, 0.1^ 
= {l+e + . . .dl+J + . . .|{1-elogn + . . .; 
u(a)| + w.t) 
{1 + e + . . .} {1 — elogn + . . .} 
+ oBj (oj^, ojo){1 — elogft + . . .] 
= ,,8/(«.)[! — eloga] + e 
o o 
om 
4&)yo,, 
a 
O), + w., 
-CO,CO. 
+ higher powers of e. 
And therefore, by the second definition of the doid^le ^ function 
C0^ + (0.2 
C,,(e, «i o.,) = oS]'(C()[l — elog«] 4- e 
Sa~ 
4a)j&)., 
— a 
-( 0 ,( 0 :, 
1^2 
{it “h f} “h W| ~j~ i ^ ^ h'g {u 4" d~ ~1~ 
— elog(a 4^4 • • • ] 
_ s' s' 
w —f 
///^=:0 v / 07 =U 
— oS/ ( a 
— 2^1 d~ ^ ~h ^’) I ^ ^ ^ 
Tims 
and 
4 2^1 "i“ ")!, 1 — ^ 4 • • • i — 14 ^ It’S "fi d" .7 
wliere fl = 4 
^.■> ^ 0 , Ct I Oj, (O.-^ - o8| 
IR 
e = (^ 
t> (e, a, I Wj, (o.,) — ,,S/ («) 
€ 
Jill '/it 
= “ S S log {(t 4 tn^oo^ 4 d“ 
/;t^=0 —0 
[over] 
