DOUBLE GAMMA FUNCTION. 
and, therefore, since x-z (^) quadratic polynomial in we must have 
+ constant. 
^3 (^) = 2 ^/ 2 ; 
P 9 / 
If we determine the constant l)y making 2 : = 0, we have tinally 
'n’ 'n rJz+—^-h~] 
(z 
"1 "3^ 
,■ = 0 
.9=0 \ 
V 9 
P' 9 ! ~ 
P - 1 
n 
,•=0 
rj - 1 
n' IN 
•S = 0 
/ rco^ SCO.-,' 
lu + PI 
t P-p,q J \ p q ' 
From the values of and m, it is readily seen that = 0, unless the axes to 
(o), + 0 J. 2 ) and [qco^ -f-include the axis of — 1, in Avliich case 
— — 1 if I(cj^ “h 1® “k and “j~ 
ju.p,^ — q- I if I(ajj + 6j^) is — VC and l(qco^ + pco.i) “k 
§ G8. The constant which enters into the transformation formula of the preceding 
paragraph can be expressed in terms of po(wj, co.^ ) and P 2 7~)- ^01’ 
pose we consider the contour integral which represents the doulile gamma function. 
Since p is a positive integer, 
1 — e“"i” 
1 _ M, n 
p- 1 
— V ^ 
r = 0 
And therefore if the integral, its contour, and the logarithm which occurs in the 
subject of integration, be defined as in Part III., we have 
J_ r C - - 2) - I {log ( - s) + ry I (/z 
27r Jl 
IL 
(1 - c-f) (1 - .-f) 
/) - 1 rj- I 
27r 
t> ( - ,:■) ' { log (-.?)+ 7} 2 2 c- 
r = 0 .•? = 0 
+ 
•! 
'L 
(1 - (1 
dz 
f 
for tlie bisector of the angle between the axes of cjjp and co.^/q is the same line as 
the bisector of the angle between the axes of l/wj and l/wo. AVe therefore have by 
§ 45, when a is positive with respect to the w’s, 
log 
"1 
CO:, \ 
a 
\ 
p 
9. / 
P3 
(Oy (0.2 
.p’ 9 
— 2^^0 ( « 
^1 ^ -2 \ / I ' \ o C< ' / ^1 ^ 2 \ n 
- , - m p,q) 2.771 —, 2/r.,, 
p (lJ ^ ‘ ‘ ~ ' V P 9 / ^ 
p- \ q-\ f'-p / r/ -k — I 1 1 4 - 1 / 
= 22 log J 3 7^ + q i \ — 2771 2 2 oSq (a + — + 
•/ = 0 s = u 
P 2 (cOy, CO.,) 
• = 0 s = 0 
-'2 '2 „S/(a + ^^+'^M2M7rr 
• = 0.? = 0 " V p 9 ! 
P 
SCO.-, ' 
,771 
SCO., 
9 
