274 
and therefore 
MR. E. tv. BARNES ON THE THEORY OF THE 
,S/)(a+ a;,)-„S/)(«) 
n ! 
= S„V)(o 1 c.,) + 
{n — A:) ! 
oS„_^.(a -f- ojj) — -7 
I Wo) 
n — k + 1 
Tims if we write 
n ! 
f{a) = gS/) («) - ^ oS,_,f (a) 
we shall have 
/ {a d- Wj) — f{c() — (o I Wo) — 
n : 
Similarly 
(7i - h + 1)! 
= 0 (“Gamma Function,” § 15). 
/’(« -r Wo) ~/{u) ~ 0. 
S'„_i.+ i (ojw^) 
and therefore since _/’(«) is an algebraical polynomial in a, it is a constant. 
On making « = 0 and rememhering that oS„(o) = 0, we obtain 
n ! 
( 7 i — Jc )! 
Ol' 
S„_, («) == {0\ 
which is the required i-esult. 
^ G. We are now in a position to prove that 
[ ('"0 “ — A* 1 2^'»+i 1 "n ^ 2 ) “h 
J (■) + 1 
^ H + O 0|«^d 
(/i + 1) . G + 2) 
['“Ml / \ 7 ^3 Cl/ /,| \ I ^ii42(^l‘^l) 
and at the same time tlie im})ortant relation 
(0 1 a.,, a.,) = ' + I “1. ‘"i)' 
Since 2 S,,(o) = 0 we see from the lundamental difference equation that 
Sk+h*^ I " 2 I 
2 ^/' (^]) — 
n + 1 
Hence, if we })ut a — co, in § 5, we see that when n—l' > 0, and /.■ > 0, 
,(o-,) 
n ! 
(7i + 1 — A) 
S'„_7- + i {o\oj.X 
Take now the relation 
