DOUBLE C4AMMA FUNCTION. 
365 
Therefore, under the same limitation, 
C“('‘+");(_ *')' jlo" ( — c) + 7} 
Now, since 
(1 - 
Vh® (« h) = S 
fJz = w). 
1 
= o(« + mcof 
it is evident that (a \ w) is homogeneous of degree -- 2 in a and a». 
Therefore, by Euler’s theorem, 
so tliat 
^ 0«) I ~ 1 
(o I oi) -h ojxp.S^ (« + oj I w) — 2i//^^“^ (« j oj) 
On integrating twice with respect to o, we ol)tain 
ccxjj^' (rt I (o) + coxji.2 j (o) = (a I ctj), 
where yi l w) is R linear function of a. 
Changing a into a + &>, and subtracting, we see that Xi 1 <^) satisfies the difference 
equation characteristic of S/ (o ] co), so that it can only differ from this function by a 
constant, wliich will vanish, as we see l)y making a = 0. 
We therefore obtain again the relation required. 
§ 72 . We proceed now to tlie analogues of Haabe’s formula. This formula may be 
written (“ Theory of the Gamma Function,” § 8) 
0) 
log (2 + *-<■ 1 w) ~ « log a~ a-\- - log 
zir 
0) 
We Avill evaluate 
f w, _ roi2 
log G (2 + a j coj, ( 02 ) dz and log G (2 + a] co.,) dz. 
The metliod wliich will be enqiloyed is the same as tliat by Avhich Eaabe’s formula 
itself Avas originally olitained ; it was, in fact, first invented for the proof of the 
present theorem. 
Let 
r"i 
/■(«) = log ro (2 + a\(x)^, w.,) dz. 
J 0 ~ 
Then 
of {a) r“i r„'(z + «) , 
*r =).n(UJV)* 
To (o + W] ) 
= log 
G («) 
— log Vi {a I oj) + log pY (ojo) + 'Idittl S/ (rt 1 0 J 2 ) , 
