DOUBLE GAMMA FUKCTION. 
285 
Thus the dilFerence of any two rational integral algebraic solutions of 
y(a + + ^- 2 ) ~ J ~ + (^-2) +/(f') = cj." 
is of the form A.a + /a. Whence the theorem in question. 
[Dr. E. W. Hobson has kindly pointed out to me that the analysis of the 
preceding paragraphs would be much simplified by starting from the direct definition 
of the double Bernoullian function in § 17. 
We should thus define the double Bernoullian function by the expansion 
1 -e- 
(1 — (1 ■ 
= "--2So(«) + • . 
ct>^a)o 
_p ^ 1^1’ *^ 2 ) q_ 
n ! 
On differentiating with respect to a, we get the expression of § 15. 
From the relation 
1 —+ l—g-az g-az 
(1—(1 — (1—(1 — e~'^zz^ 1 — e~“a-’ 
we obtain the fundamental difference relations for the double Bernoullian function. 
The result of § 10 follows from the identity 
\—g-az 1 — g{oii + u.^-a)z ^ 
(1 — ) (1 — ) (1 — (1 — 1 — (1 —^ 
and that of § 14 from 
g-aqi _ 
a>|Z 
(l-H"? )(i_e“T) 
Inasmuch as theoretically the properties of an algebraical polynomial should not 
be derived from consideration of the coefficients of an infinite series, the original 
investigation has been retained. I had already j^roposed to myself to work out the 
theory of multiple Benoullian functions by a method closely allied to that suggested 
by Dr. Hobson. — Note added Jidij 3, 1900.] 
P-] 2-1 
N N 
X;=o ;=o 
1 — e-i“+ k + 
'?)4 = 
PI 
Part H. 
The Double Gamma Function ro{a | coi,<yo) cmd its Fdememtarxj Properties. 
§ 18. In the elementary consideration of the simple gamma function it was found 
to be necessary to rely on two algebraical limit theorems :— 
(1) Euler’s theorem Lf[l + ^ + . . . + — — logn] = y. 
(2) Stirling’s theorem Gt 
71 ! 
yjn + ig-B 
= A,/27r. 
