DOUBLE OAMMA FUNCTION. 
317 
Hence we have 
tUd - s) Ce — + l)tO| + Ct] 3 _|_ g [(Q/l + 1)10^ + ft] ' 
_ tUd - s) r 
27r j 
1 
(1 — (1 — e 
(— zV ^dz 
(s — 1) (s — 2)a)^Wo 
(/picod" " 
+ 
1 2(f -j- co^ "L (jo.2 r 1 
- 1 ■ 
(OjWo 
+ 
1 
^ :r+ ’ 
dinw^y M s — 1 ^ ^ 
2>Sh(^f A ohj (ft + fO.i) 
{pnwd' 
i) J 
_l As + HI- — 1\ J nS,„(ft + (Mj) + I 2^1,1 (ft- + Wo) + 
J/d.= 1 
M + S 
/ L {pnco^y'^'^^ 
In an exactly similar manner we find, since 
2^M (ft 4* W] + Wo) = fi- Wj + ojo) fi- 
that 
fP f' s) r g~[«+(i)n +l)ui+ (<2/1+Ijwa]- 
){ fj-la+Qiy 
J (I - e"" 
27r J (I — e “i’) (1 — e 
1 ] 
(-zy-h/z 
2(1 A coi A Wo 
(s — l)(s — 2)(yi&)o (p/i&)j A Qfico^y " (s — l)2(yj&)o A piw^) 
,^-i 
■ nh]^ (ft A &)o A Wo) I ^ / \m [^ A Wj A Wo) A oB,„^.j 
( jrno)., A qnco^y m=i 
m 
^ \ I \ ^0/ I ■ 
j ( pnw^ A qnoydf 
If now we group together all the results which have been obtained, we find the 
asymptotic quality, true for all values of s, and for values of «, w^, and wo, whose real 
parts are positive. 
im q.l 
V V 
ti,=0 m,=0 (ft A HqWj + ^^qWo)' 
- ^o(<S, ft I O^i, Wo) 
+ 
1 
1 
1 
in A w, A Wo 
(s — l)(s — 2)ft)^&)o [(pHWj A qnw^y ^ (p/tw,)^ 
1 1 I 
J. 
{qnw2y~''‘ 
2(s A l)wiWo [(p«Wi A qnfo^y ^ (pHwd’-' iqnoy.^y ^ 
s — 1 i_w^(p?iwd* ^ ^ i 
+ 
2 ^A(ft A Wj A Wo) o&i (ft A Wj) 2^1 (® A Wo) 
{pna>-^ A qnoy^y 
(FIWj)" 
{qrmyy 
I (— )r fill s — Id . ^,n{n A w^ A Wq) A ^B 
m=l \ A (i>Wl A ^Wo)”^' 
'« + l 2^)n(ft A Wj) + 2®'" + ! 
(pwP"+” 
ohmCft A Wo) A oB'^ + l 
( 2 ^ 3 ) 
ni-Vs 
(A) 
It may readily he seen, just as for the case of a single parameter, that the series 
proceeding by powers of ^ is a series of powers of a real variable, wliose line of con- 
vergency is of zero length. 
