470 
ON THE BINOMIAL THEOREM, 
[533 
S 3 
A product 
on (on — a) ... [to — (r — 1) a} 
can of course be expressed in the factorial notation, viz. it is 
and on this account it is not in general necessary to employ a notation such as 
[to, a] r to denote such a factorial wherein the difference of the successive factors instead 
of being = — 1 is = — a; in particular cases where factorials of the kind in question 
are used, it may be convenient to employ such a notation. In particular it is some 
times convenient to use the notation [on, — l] r or better [m\ r to denote the product 
m(m + 1) ... (on + r — 1), 
where the successive factors instead of being diminished, are increased by unity. It 
may be noticed, that reversing in this last product the order of the factors, we find 
[m) T = [to + r — l] r ; 
a somewhat similar formula, but employing only the ordinary factorial notation, is 
obtained from the equation 
[to] 1- = ooi (to — 1) ... (to — r + 1), 
by first changing the sign of on and then reversing the order of the factors; viz. we have 
[— to]’’ = (—) r [to + o' — l] r . 
Reverting to the process used for the development of the expressions ( ) 9 , where there 
are two columns, the one of greek, the other of latin letters; it is to be remarked 
that although the order in which the successive lines are evolved is not material for 
the purpose of the theorem, yet that a certain definite order of evolution has been 
made use of; thus in regard to J’ ^ J , the column of greek letters, giving 
the homogeneous products of the second order in (a, /3, 7, S), was 
a 
a 
a 
(3 
a 
7 
/3 
/3 
a 
8 
¡3 
7 
!3 
3 
1 
7 
1 
8