416 On a new Method of treating 
(m+ 1)re il i ms 4) (mp1)! mnt} 
yet 1"|! x (n+ 1) a ye+ih 
{1 
(m+1)" 
yt 
Proposition 2.—In any two consecutive orders of figurate 
numbers, the sum of m terms of the antecedent order is equal to 
the mth term of the consequent order. 
Let 1, 0, c,d, &c. 
1, B, y, 2, &e. 
be any two consecutive orders of figurate numbers ; than by the 
last proposition 
The sum of these equations is 
1+6+y+b4+c+d=6+y4+2; 
take away the common quantities 8, y, and there will remain 
ltb+c+d=8; Q.5.D. 
Coroliary 1.—Hence the sum of m terms of any order is 
equal to the mth term of that order, when the exponent of each 
of its terms is increased by unity. 
Corollary 2.—Hence the first order of figurate numbers being 
given, the consecutive orders may be derived to any order re- 
quired ; thus, ; 
lst ‘order: fae) 3 4 aA 
2d order 1 ¢& 6. 18 Loe ae 
38d order 1 4 10 20 35 56 
Ath order 1 5 15 385. 70 126 
&c. &e. 
Proposition 3.—The mth term of the nth order is equal to 
the (2+ 1)th term of the (x—1)th order. 
n\i 
For 
1"\ i 
bers. Now the factorial m"!! may be resolved into two factorial 
factors, so that one of them may have the given exponent m—1 
(see factorials): therefore ml! =m" —™ FN x (ny Tb ison also 
the factorial 1"!! may be resolved into two factors, so that one of 
them may have the given exponent m—m-+]1, therefore 
n{l__ ym—I]1 n—m-+1|1, 
ya) "xm UN; 
is the mth term of the mth order of figurate num- 
en) 1 at | i 
ie ml} am m+1| x (n+-1)” jl ty (n+1)” |i : 
ee yr ase m—1j1 n—m-+ 1]1 m—I{1 
1 x m 1 
Corol- 
