414 On a new Method of treating 
iI 
mm‘ = m be the one factorial factor, then will (m+ i eta be 
the other. Or if the one factorial rastes be m"—'I". then will the 
other factorial factor be (m+n— 1)'! lomtenee ly 
4, Resolve m”|¢ into two factorial factors, so that one of them 
may have the given exponent 7. 
By rule, x—r will be the exponent of the other. Therefore if 
ml be the 
be the one factorial factor, then will (m—rc)"—"!° 
other, ; 
Or if m"—"l° be the one factorial factor, then will [m— 
(n—r)o}"° be the other. 
5. Resolve m"'! into two factorial factors, so that one of them 
may have the given exponent 1. 
By rule, n—1 i is the exponent of the other. Therefore if m - 
=m be he one factorial factor, the other will be (m— yet. 
Or, if the one factorial factor be mt 
(m—n+1)'l! 
m”!. 
» the other will be 
=m—n-+1, which is the last term of the factorial 
THEORY OF FIGURATE NUMBERS. 
ily bg! Qi VERS 
Def. 1. In any number of series < 7? b,5 6, d,, &e. 
bettgg bys)! Chg pain Bate 
&e. &c. &e. 
placed in due order, if 2 be the number of any series begin- 
ning with that which is placed first, and m the number of the 
term in the mth series, and if the mth term of the mth series be 
njl 
expressed by aos each series is called an order of figurate 
1 
numbers *, 
Corollary 1.—Hence by this definition the first order of figu- 
rate numbers will be the series of natural numbers 1], 2, 3, &c.: 
: ml} 
for if in 
yl 
we make m successively equal to 1, 2,3, &c. and 
me gl 3) {1 
pi’ yur 
same as the numbers 1, 2, 3, &c. 
m equal to 1, we shall have ——&c., which are the 
pur yt 
* The author has here adopted Legendre's definition of figurate a Deh 
Def. 
