468 
ON THE BINOMIAL THEOREM, 
[533 
For 
a, /3 
a, b, c, d, e)i 
a, 
a, 
a 
a 
a, 
a, 
/3 
a 
a, 
a, 
/3, 
/3 
a 
P, 
/3, 
/3 
a 
/3, 
f3, 
A 
/3 
b 
viz. the expression is 
(a — a) (a - b) (a — c) (a — d) ... + (/3 — b) (/3 - c) (/3 - d) (/3 - e). 
Finally for 
( a 7 . | write a, a, a, a, a \ a, b, c, d, e\ 
\a, b, c, d, el 5 
viz. the expression is 
(a — a) (a — b) (a — c) (a — d) (a — e), 
which explains the law of the formation of the several coefficients. It is to be observed 
that in forming the development of any symbol, for instance f a> f’ ^ 1 > the first 
\a, o, c, ct, ej 3 
column contains the homogeneous products, 3 together, of a, /3, 7; the second column 
the combinations (that is, combinations without repetitions) 3 together of a, b, c, d, e : 
the top line is a, a, a | a, b, c and to form the subsequent lines we must for any 
advance a into /3, &c. of a greek letter make the like advance a into b, b into c, or 
c into d, of the corresponding latin letter. 
Two particular cases of the theorem may be noticed : if the latin letters all vanish, 
we have, for example, 
h 5 = (h — a) (h — /3) (h — 7) (h — 8) (h — e). 
+ (h — a) (h — /3) (h — 7) (h - 8) 
+ (h — a)(h — P) (h - 7) 
+ (h - a) (h - p) 
+ (h — a) 
+ 1 
• H 1 (a, /3, 7, S, e) 
• /3, 7, 8) 
• H z (a, /3, 7) 
.# 4 (a, /3) 
• H 5 (a), 
where the symbols H denote the sum of the homogeneous products of the annexed 
letters, taken together according to the suffix number : the last coefficient H 5 (a) is of 
course = a 5 . And if the greek letters all vanish, then we have in like manner 
(.h — a) (h — b) (h — c) (h — d) (h — e) = It 5 
— /i 4 Ci (a, b, c, d, e) 
+ h 3 C 2 (a, b, c, d, e) 
— h?C 3 (a, b, c, d, e) 
+ hC 4 (a, b, c, d, e) 
— C 5 (a, b, c, d, e),