348 
MR. E. 4V. BARNES ON THE THEORY OF THE 
Thus 
1^2 («) - 
— a 
Wl + C0.3 
2(/7/. + m) TTl 
exp. 
U 
«= 0 
fo {■‘<,a) - 08 / (ft) 
I •! I g-An'") 2 (/,i -i- in') 7 
P 2 ^ 2 ) 
We may utilise this formula to establish the fimdamental difference equations for 
the double gamma function. 
By the definition of the doidde (, function of § 57 
C: {('- + <^i) ~ ^2 
[u -f- + ['ll -f" l] COq) — [o + (ll + l) Wo] 
/i 
V 
1 
ii,, = 0 (« + 
Tlierefore, in the limit when n is infinite, 
U 
= 0 
^■2 ('I + Wj) — 2*^1 (C' -f- &)j) ^3(^0 — 381^^) 
S S 
— N ~h —■ oh*[ [u ~h Wj -(- [11 -}- I j Wo] log’ -|- Wj -|- [ll -|- 1 ) Wo 
. = 0 
d- oSj [o. + [n +1) Wo] log [« + (/! + 1) Wo] + — [a + (;/ + 1) t^o] 
a Wo -Wo 
On reduction we obtain 
W]^ + 03 .0 
:Wo 
1 J To(" + Wl) --3 
2 (jR + )/d; TTt Si'(« 1 OJ.,) 
= U 
^ log [a + uJoWo) — (— + b' + ^2 _ ^ I _|_ 
hl.-i = 0 
(O:, 
This latter expression is, l)y the expansion oljtaiued in the “Theory of the Gamma 
Function,” § 30, equal to 
, Fj (c. I W 2 ) o' o ' / I \ 
— Joy — 2m7nDi («. w.,). 
Pifwo) 1 V I -/ 
[The term '2'in'Tn[7i + 1) which arises is absorbed by the identities w'hich change 
log [a + i^oWj) into log 7)),ow., + log ^1 + -— ^J. The prescription of the absolute 
logai’ithms has been tbroughout left indeterminate.] 
We thus have 
lo + W)) _ I l O^ I ^*^ 3 ) -2,,mS,'(ffi I M,) 
(^) P\ 
one of the fundamental formulae of 5 23. 
Sufficient indication has perliaps now l:)een given of the alternative development of 
the theory of the double C function. 
