Factorials and Figurate Numlers. 415 
Def. 2.—The mth term of the first, second, third, &ec. order 
is called the mth vertical column. 
i: Sir 23h 
Thus Fane m ,™  &c. is the mth vertical column. 
] ifl 121! yl 
Corollary.—Hence the nth term of the mth vertical column 
is the same as the mth term of the nth order of figurate num- 
ml! 
I 1 
and by the last definition the very same is the mth term of the 
mth vertical column. 
Def. 3.—The first term of the nth order, the second term of 
the (n—1)th order, the third term of the (x—2)th order, &c. is 
called the mth diagonal series. 
yr gn—I{k gr—2|1 
i ye you yl! 
bers; for by definition 1, 
is the mth term of the mth order, 
, &c. is the mth diagonal series. 
Corollary 1.—Hence if x be the number of the term of a dia- 
n=x+1{1 
az 
2 where x must 
gonal series, then any term will be 
y" 72+ 1| 
never exceed 2+ 1. 
Corollary 2.—Hence if m be made equal to 1, and x suc- 
cessively equal to | and 2, the first diagonal series will be 
yi-l+l  gl—24+1)1 Ut gol 
ee Ss thatcis, >» —.., which in effect is the 
qi-itelil yl—2+11 pub 70/1 : 
same as l, l. 
Corollary 3.—Hence the mth terms of any two consecutive 
orders of figurate numbers will also be the mth terms of two con- 
secutive diagonal series. 
Proposition i.—The (m+ 1)th term of the (w+ 1)th order is 
equal to the mth term of the (7+ 1)th order and the (m+ 1)th 
term of the mth order, 
For (m+ mem 
petit 
is the (m-+1)th term of tlie (n+ 1)th order. 
n+1}1 
Now each of the terms of the fraction (m+ 1) 
peril pid he 
resolved into two factorial factors, so that one of them may have 
the given exponent 1 : therefore the factorial (m+ 1)"*"!" js equal 
to (m+1)"!! x [m+(n+ 1)}=m"* eh +(n+1) (m+41)"!', and 
the factorial 1"!!! =1"l! x (n4+1): whence 
(m+ 1) 
