282 
MR. E. M. BARNES ON THE THEORY OF THE 
In tliis expansion the double Bernoullian functions have and Wo for parameters, 
and the expansion is valid provided 1 2 | is less than the smaller of the two quantities 
2771 j 
2771 
"1 ! ’ 
&’2 
For Avithin a circle A^drose radius is less than this quantity, the function 
z-e 
(1 — c “1^ (1 ” ^ 
has no poles, and hence it is expansible in a Taylor’s series of poAAmrs of 2. 
Thus Ave may assume, for all finite Amines of a, 
Ay«) 
— AQ(a) + 
/iFO 
1! 
+ ...+(-) 
>;-1 /»fo) 
71 ; 
+ 
(1 — e “‘"J (1 — “ 2 --) ^ 
where it is obAuous that the functions of a Avhich enter as coefficients are all alge¬ 
braical jDolynomials. 
Change noAv a into a -|- and subtract the expansion so obtained from the one 
just written. 
We find 
re--- Aq«)-Aqa + a)d w,a , , a 1 /ifo)+ ‘“i) .. . 
- = -;-Ao(«) -f Ao(a + wi) -1-7A- 2 + 
1 : 
„_1 + «l) 
71 i 
+ 
' But in the “ Theory of the Gamma Function,” § 20, Ave obtained the exj^ansion which 
may be written 
■ Tl = - 1 j 2 " + ... 
1 - c- 
Equating coefficients in these tAAm ex^Aansions, Ave obtain 
Ai(^ ffi ^ 1 ) — A]^(nj 0 
A|,(a + ojj) — A^,(«) = S/~^(rt|a)o) 
/l(« + - /l(«) = S'i(rtla>o) 
f„{a + Wj) —fn{ci) = S'„(a| Wo) 
and it is obvious that a similar set of equations hold in which and are inter¬ 
changed. 
Hence Aj(n) is a constant AAdiose Amine, from the first term of the expansion, is 
(O^CO:^ 
= oSi®(o,[a;i,a;.) by § 4. 
Again A,j(a) = for these two expressions can only difter by a 
constaait Avhich 1jy § 4 and the actual expansion is at once seen to A-anish. 
