DOUBLE GAMMA FUNCTION. 
369 
Now it has been seen that (“ Theory of the G Function,” § 17) 
2 * 61 ? 
A = 
vr 
and therefore 
'^IC; 
A —— (eoj)-', 
Pi(«) 
which is the relation which was used at the end of the preceding joaragraph. 
S 75. It is interestino- as a verification of the alo-ebra to notice that Alexeiewsky’s 
0 o o 
analogue of Faabe’s formula (“ Theory of the G Function,” § 16) yields the result 
of § 73 in the case where the parameters are equal to one another. 
This theorem is expressed by the formula 
[ log G (a; + 1) fA = I log G (i) + Af log TT + 3 ^ log 2 - Ar, 
and therefore, utilising the relation between the G and double gamma functions. 
18 
2a) 
log To (« + w j co) da = - y log G (^) + ^ log Stt + ^ log 2 + log w. 
If now we ex])ress log G ( 5 ) in terms of log (w) by the formula of § 74 , 
we find 
P • (^) 
log To {a + 0 ) ^a>) da = o) log p.^ (a» ) + OJ log 
JO 
+ 
CO 
Pi(«) ' 12’ 
a formula which is equivalent to the result of § 73. 
§ 76. By comhining the results of §§72 and 73, we may now write down the 
value of 
log 1 0 a I coy, ^i) • 
For we liave seen in § 72 that this integral is equal to 
ra r(ji, 
— I log Ifi (a I &).,) + a log p^ (ojo) + 2 rinn (a | oj^) + log U, [z \ oio) dz, 
•0 ~ “ ". 0 ~ 
which expressioii in turn is equal to 
— ft log (ft I oj^) “h (ft I 6 J. 1 ) 1^1 -)- 2?)i7riJ — coo log (ft “h Wo | Wo) A (ft "h Wg) log pj (^ 3 ) 
+ Wj [logpo (^ 1 , OJ.^) — (m + ni') 277-6 „S/(o)] + *^3 log “1“ fiUl A- Stotti). 
Pi (" 3 ) 
Thus 
ru>i 
l0gr2(z + «1 Wi, Wo)d2 
J 0 
— ftlog ^^^'^^^ ;y + COy [log p., (ojj, OJo) — (m + ni) 27rtoS/(o)] 
Pi (" 2 ) 
Wo 1 
+ (1 + 2 m 7 ri) <|Si(«|&) 2 )+Ylj — W^log 
3 B 
G + <^ 2 1 ^ 2 ) 
" 2 ) 
VOL. CXCVI--A. 
