DOUBLE GAMMA FUNCTION. 
277 
On differentiation we have 
{a I a», oj) — S,, (a I oj) + - I ~ (a ] oj) -f- (<y)) 
0 ) 
CO 
CO 
so that 
jS',, (o I OJ, w) = — Sh (o I w)- + i (oj), 
and therefore oB„ (co, co) = — j^B,, (co)- iB«+i ( co), 
cv 
§ 9. We now see at once that 
oS„ (a I co^, CO.,) ■ 
■,11+3 
CO I + CO.2 
(w F 1) F 2) coj C 0.2 2 (n F 1) coj^ co^ 
- oBj^ n'* 
H~ ( 1 ) 3^2 (<^1, £^3) ft" ^ + ( ] .2B3 (w^, C0.2) cG ^ + . 
pnid so com]dete tire expansion of § 4. 
For by Maclaurin’s theorem we have 
3S„(al»,.<».) = a.S',(o) + ^~L‘‘‘ + . . ■ 
since the hijxher differentials vanish. 
O 
From the few terms found in § 4 we see that 
,s,S"*Uo) = 
01 ! 
Now when n > k and k > 0 , we have 
CO^CO.2 
oil (CO^ F cot) 
2co-^co2 
3^/^-U](f^n "3)> 
and thus we have the expansion in question. 
§ 10 . It may now he shoAvn that 
(-) 
)i — \ 
,S„(«) = ( —)" oS„(w 3 + oq — O.) + q ^ . [^«+i + i (tt|£F)]> 
ih “i“ F 
or, as we may write it, 
oS,, («) = (~)"3S,, (w^ -F on — a) 4- ( —[i^F + i("l) + iE>« + i(oq)]. 
Ptemembering the value of, iB„+i(a>) we thus prove that 
gS;, {a) = oS,, (cui + wj — a), when n is even. 
Tc - 1 
3 S;, (a) 
oS„ (oj, + Wo — «■) + ^ Ba + 1 (w^" + w/), when n is odd. 
\ i I .V / > ri, F 1 "V- 
