412 On a new Method of treating 
mated, leaving a very small place for the incisores, which are very 
small, very short and flat, the two lateral ones on each side are 
situated diagonally ; the second behind, and the two middle ones 
are only half'the size of the others. The tail is bushy, particu- 
larly at the top,-where there is a white pencil of long hairs; the 
brown of the remainder is darker than on the body. 
From the above accurate description it will appear evident 
that’ this animal is very different from the common marten of 
North America. It must be a very ferocious little animal, which 
is indicated by the strength of the teeth. ‘ 
LXVIII. On a new Method of treating Factorials and Figurate 
Numbers. By Mr. Peter NicHoLson. 
To Mr. Tilloch. 
Sir, — As the following method of treating factorials and figu- 
rate numbers is new, I hope you will have the goodness to in- 
‘sert it in your valuable publication The Philosophical Magazine, 
as it will be found to apply to many of the most useful parts of 
algebra; as in the binomial theorem, in equations of all di- 
mensions, in combinations, &c. 
London-street, May 17, 1819. PeTER NICHOLSON. 
FACTORIALS. 
Definition. —An algebraic product of which the difference 
‘between every two adjacent factors is equal to the same given 
number, is called a factorial. 
Notation.—In a factorial are to be considered the number of 
factors, otherwise called the exponent, the first factor, and the 
common difference, whether + or —. 
Let m be the first factor, 2 the number of factors, and ¢ the 
common difference ; then every factorial may be thus indicated 
m"!’: let n=4 and c=1, then will -m"!° =m'*!! = m(m+1) 
(+2) (m+3). Again, if n=5 and c= —1, then will m"l°= 
m?\ =m(m—1) (m—2) (m—3) (m—4). Again, let m= —p, 
and c= —e ; then will mre (—p)"!*, which will be. affirmative 
or negative, according as 7 is even erodd. Thus let p=3, n=4, 
and e=2; then (—3)*I?=(—3) (+5) (—7) (—9)=945. Again 
let n=5 5 then (—3)*!2—(~3) (5) (—7) (—9) (—1]) = = 
Jee m5 5 then (—8)""=(—8) (=5) (~7) (=9) (—1) 
_ Proposition.—Any two factorials in which the base of the one 
is equa] to the suin formed by adding the product of the expo- 
nent 
