DOUBLE GAMxMA FUNCTION. 
851 
- w — — y.2j (<^1, OJ.,) + Iug; + [log w, + log — log (oj| -f" OJ,.)] 
' ' (0^0).^ 0)^(0.^ 
n n ^ 
/n^ = 0 9/10 = 0 -- 
the principal values of the logarithms being taken. 
Therefore 
so that we have 
I- = — — Ion m, 
/,=o 2=0 "" \ ; 
ni- 
Ion rn. 
C? 
Integrate again with respect to 2 and we find 
pco-^ + '/Ct)o'\ V/Us 
nl — 1 //I— 1 / 
mxjj. 2 ^^'’(m z) = S S + 
^) = 0 7 = 0 "" ' 
log in . . (ii.) 
where s is constant with respect to 
Now we have, in the limit when n is infinite, 
i/;T> (s) = — -721 (wo Wo) — yoo (w,, Wo) — 7 — S S' 
0 0 
■ 1 
r + n 
o 
Hence we find, on equating the irresolnble terms involving in the same wav oi 
both sides of the identity (ii.), that, in the limit when n is infinite. 
r Ill'll-^ //i—1 1 
in |S S' ^ ~ yo., (w^, Wo) !> = s + nr ^ ^ 
nil 
0 
/til ^‘^2 
- 7id 
Wi, Wo 
-v -4- V V' — 
Til ' 02 
(I 
4" / /2Wi + gw7 , 
0 
/' ‘1 
ill 
and, therefore, when ii is infinite, 
.s‘ = m i N i' — — yoo (w^, ojo 
/ii/i+//i—I 1 1 r 1 
^ N"/-U,.. .,.A I — js S' - — yoo ( 
W|, CO.) 
+ 9/i (ni — 1 ) -^-7 -■ jS S' j^o + y^i (wi, Wo 
Now in § 23 we have seen that 
S S' - — yoo = d) lc)g w + (?2 + 1 ) (77 + ) l*''g' ("1 + ".i) — I'^g 
u 
rn, nu 
2w^Wo 
1 , 'ImiTL 'Im'iTL 
-log Wo — - — - 
W, Wo 
LX'"! 
w 
Wo 
w 
““ 9 . + " 2 ) “■ l<^g "1 “ ^'^g 
2W|Wo 
and, therefore, after some reduction, we see that 
