Factorials and Figurate Numlers. : 417 
Corollary 1.—Hence one expression of figurate numbers can 
easily be converted into another equivalent expression by the fol- 
lowing. 
Rule.—Add unity to the exponent of the numerator of the given 
expression, and it will give the first factor of the numerator of 
the new expression; and take unity from the first factor of the 
numerator, and the remainder will be the exponent of both the 
numerator and denominator of the new expression; the first 
factor of the denominator being the same as that of the given 
expression. 
Proposition 4.—The first, second, third, &c. terms of the mth 
- yertical column are equivalent to the second, third, fourth, &c. 
terms of the (m—1)th order of figurate numbers. 
For by definition the first, second, third, fourth, &c. terms of 
m—\|1 
the (m—1)th order of figurate numbers are respectively TG 
gmk, gm ill qm— i 
ee | ee: then by the rule: to, corollary 
ye “ym ll yi 
have the equivalent series —_, ——, —_, 
jolt pit yi ysl 
} matt 
definition 2, the second, third, fourth, &c. terms —_, —_, 
pili yal 
31 
one &c. are called the first, second, third, &c. terms of the mth 
1 
vertical column. Q.£. D. 
Corollary.—Hence because by proposition 2, the sum of m 
terms of the preceding order of any two consecutive orders of 
figurate numbers is equal to the mth term of the consequent or- 
der; the sum of m terms of the (m—1)th vertical column must 
be less by unity than the sum of m terms of the (m—1)th order 
of figurate numbers. 
Proposition 5.—The nth diagonal series of figurate numbers 
ial 
- de het J y2it? rele 
For by corollary to definition third, any term of the mth dia- 
n—ax+I}1 
gonal series is —___—_.. 
yrret i}I 
Vol. 53. No. 254. June 1819. Dd But 
