[119 
119] ON A THEOREM FOR THE DEVELOPMENT OF A FACTORIAL, 
99 
which may be thus written, 
FACTORIAL. 
L85.] 
implicity of its 
x — c... in the 
c. 
factor the term 
the term a — a, 
b\ 
i the right-hand 
— a + a — c; the 
5 ~ /3 x - 7 
v — /3 
(x — a)(x — b) (x — c) = 
(a? - a) 0 - /3) 0 - 7) + 
a, 
_a, 
/3, 
b, 
7 
cJi 
x — a x — /3 + 
a, /3, 
r* 
_a, b, c_ 
! ‘ c -“ + L b, o\ 
Consider, for instance, 
— a — a a — b + a — a ¡3 — c + j3 — b /3 — c; 
2 
a, /3, 
a. 6. c 
then, paying attention in the first instance to the Greek letters only, it is clear that 
the terms on the second side contain the combinations two and two, with repetitions, 
of the Greek letters a, /3, and these letters appear in each term in the alphabetical 
order. Each such combination may therefore be considered as derived from the primitive 
combination a, ot by a change of one or both of the a’s into ¡3; and if we take 
(instead of the mere combination a, a) the complete first term a — a a — b, and 
simultaneously with the change of the a of either of the factors into /3 make a similar 
change in the Latin letter of the factor, we derive from the first term the other terms 
of the expression on the right-hand side of the expression. It is proper also to 
remark, that, paying attention to the Latin letters only, the different terms contain 
all the combinations two and two, without repetitions, of the letters a, b, c. The same 
reasoning will show that 
x — ax — bx — cx — d = 
x — a x — f3 x — y x — 8 
+ 
+ 
+ 
+ 
/3, 
7) 
8~ 
-a, 
b, 
c, 
d\ 
1 
a, 
¡3, 
7» 
X 
La, 
b, 
c, 
d\ 
2 
a, 
/3 
X 
La, 
b, 
c> 
cL 
3 
a 
-a, 
b, 
C, 
d_ 
> 
4 
x — a x — ¡3 x — y 
x — a x — (3 
x — a 
where, for instance, 
a, /3 = (a —a)(a—b)(a — c) 
_a, b, c, cLl 3 +(a — a) (a — b) (/3 — d) 
+ (a — a) (/3 — c) Q3 - d) 
+ (/3 - 6) (/3 - c) (/3 - d), &c. 
13—2