98
[119
119.
ON A THEOREM FOR THE DEVELOPMENT OF A FACTORIAL.
[From the Philosophical Magazine, vol. VI. (1853), pp. 182—185.]
The theorem to which I refer is remarkable for the extreme simplicity of its
demonstration. Let it be required to expand the factorial x — a x — b x — c... in the
form
x — a x — ¡3 x — y...+ Bx — cl x — /3.., + Gx — a... + D ... &c.
We have first
x — a = x — a + a — a;
multiply the two sides of this by x—b\ but in multiplying by this factor the term
x — a, write the factor in the form x — $ + /3 — h ; and in multiplying the term a — a,
write the factor in the form x — oc + a — h; the result is obviously
x — a x — b = x — a x — /3 + 0 — a + /3 — b) x — a + a — a a — b \
multiply this by x — c, this factor being in multiplying the quantity on the right-hand
side written successively under the forms x — y -t- y — c, x— /3 + /3 — c, a — a + a — c; the
result is
x — a x — b x — c = x— ax — /3 x — 7
+ (a — a + /3 —b + y— c) x — a x — /3
+ (a — a a — 6+a — a/8 — c + /3 — b ¡3 — c) x — a
+ a — a a — b a — c,
