300 
Mil. E. W. BAENES ON THE THEORY OF THE 
Hence, making t = 0, we have 
1 
v/(27r/«j) 
which is one of the relations rec[nired. 
We thus see that ro(&J2) independent of and not only so, hnt its value is 
snhstantially a cpiantity that appeared several times in the theory of the simple 
s:amma function. 
o 
Thus we saw [§ 4 cor. “ Gamma Function ”] that 
that (§8) 
71—1 
n 
(v 
/( 
27 r/a) 
f log Tj{z\o))dz = 0) log v/( 27 r» ; 
•' 0 
and the most general form of Stirling’s theorem ^vas seen to Ije 
pn 
log 11 [a + m^co) = 2)^1 n log n — n] + 7i {Si®(rt -j- a))poj logpwl 
jji, = 0 
+ ■; 1 + s^'{a)] log 72 - logr2(«) + log v/( 27 r» 
^ 1 37?7(«. + w) T 
+ S/{a + fo))logpctj + 
V 
VI = 1 
W +1 
7 nn" 
(pcoY 
We now see an additional reason why it was proposed to write 
Pi(w) = ^{'Itt/co) , 
and to call 7n(t^) fii^d pi{o}) the two simple girinma modular forms, the latter being 
sometimes called the simple Stirling modidar form. We sliall see that there exist 
tliree double gamma modular forms 
73 i("n "2)5 722(^0 ^^2) pe("n ^^2) 
of exactly analogous nature. 
§ 26 . We proceed now to connect tlie function ^(2; | Wi, w,-.) with iUexeiewsky’s 
function G(2|r), some of whose properties were investigated in “ The Genesis of the 
Double Gamma Functions.” 
In the first place, we take t == and then we have 
and wherein 
ft — h. 
p 102 ^<^2 
' . n n 
07.2 = 0 ^ ^ 
1 
where H = 7?2^<y^ + 7)ioaj„, 
'j- 
a = y log 2ttt + ^ log T — yT — C(r) , 
h = — rlogr-- t'D(t). 
