466 ON THE BINOMIAL THEOREM, [533 
But the method of varied multiplication may be applied to the demonstration of 
a much more general theorem ; viz. we may use it to develope a product such as 
(A — a) (A - A) (h — c) (h — d) (h — e), 
according to a series of products 
(A — a) (A - /3) (A — 7) (A - A) (A - e), 
(h - a) (A - /3) (h - 7) (A - $), 
(h - a) (h - /3) (h - 7), 
(A - a) (h - /3), 
(h - a), 
1. 
For this purpose, starting with 
h — a = h — a + a — a, 
we multiply by h — b, written first under the form h — ¡3 + /3 — b, and then under the 
form h — a. + (a — b) ; we have thus 
(h — a)(h — b) = (h — a) (h — /3) 
+ (A- a) {(a -a)-f (/3 — A)} 
+ 1 (a — a) (a — b), 
and so on. It is easy to see that we may for instance write 
(A — a) (A — b) (A — c) (A — d) (A — e) 
= (A - a) (A — /3) (A — 7) (A — S) (A — e) 
4- (A — a) (A — /3) (A — 7) (A — 8) I 
+ (A - a) (A - /3) (A - 7) 
+ (A — a) (A — /3) 
+ A — a 
+ 1 
a, ft 7, 3, 
.a, b, c, d, 
'a, ft 7, 3 
.a, b, c, d, 
ft 7 
.a, A, c, ft 
' a , £ 
A, c, ft 
[a, b, c, ft