AIS Ona new Method of treating Figurate Numbers. 
n—x+I]1 n—2+ IL 
n 
But by the principles of factorials AEE = eee ; 
where the first factor in the numerator of the second side of the 
equation is the last factor of the numerator in the first side; and 
because the common difference of the factors of the first side is 
+1, the common difference of the factors in the second side 
must be —1; also because the reverting of the fraction does, not 
change the number of factors, therefore the exponent of the nu- 
merator must be the same on both sides, as is exhibited. 
eel el xx —2r7+2}1 atoll 
Now, $$. Sr 
ypaaart fl yet x ree Fa yeh 
Let x be expounded by 1, 2, 3, &c. in the last side of this 
f nel. All 
equation, and we shall have 1, 7, —_, —_, &c. QE. D. 
yl ysl 
Proposition 6.—The sum of any two consecutive terms x and 
x+l1 of the mth diagonal series of figurate numbers is equal to 
the (x+1)th term of (n+1)th diagonal series. For, by Pro- 
position 1, the sum of the mth term of the (z+ 1)th order, and 
the (m+ 1)th term of the wth order is equal to the (m+ 1)th 
term of the (w+1)th order. Now the mth term of the (n-+1)th 
order and the (m-+1)th term of the 2th order are any two con- 
secutive terms x and x2+1 of any diagonal series 2; also the 
(m+1)th term of the (w+ 1)th order is the (v+-1)th term in 
the next diagonal series following. @. E. D. 
Corollary |.—Hence if any diagonal series be given, the next 
following will be found: Thus, let 1, B, C, D, &c. be given, then 
the next will be 1, (1+B), (B+C), (C+D), &c. 
Corollary 2.—Hence if the first diagonal series be given, we 
may derive as many consecutive diagonal series as we please, as 
in the following table : 
First diagonal series 1 
Second diagonal series 1 l 
Third diagonal series | tis Sen Hs | 
Fourth diagonal series | iia: Baia! iy: 
Fifth diagonal series ‘1; 3 
&c. &e. &e. 
“ 
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Ne 
7) 
LXIX. Re- 
