of a Series in the inverse Method of finite Differences. 127 
= ( by theorems for figurate numbers ) 
/«4-2\/w-f3\ I n + in \ 1 m+i , / , \ 
-n[— J(-r]--(£^.) A £. +”(»+ 1 ) 
» + 3) + | » + m j 
71 (n -\- 1 )(?2-j-2) |^~ )• ' ,2 + 3 -f-&c. (to m terms 
and m factors in each term ) . 
Cor. 1. When n—\ the coefficient of h m = 
77 z ( —f- 1 ) A i wz-j-i 1 (m — i)»«(m+i) A e n m±z[m— 2)[m—i]m[m + i) 
0 -T- 0 ‘ — — 
_ » 1.2.3 _ 1.2. 3.4 
1.2 
A 3 o m+3 &c. 
(to m terms). 
Cor. 2. The same investigation holds for the coefficient of 
li n in the expansion of + &c.]”. So that substi- 
tuting — n for n in the above expression we obtain by Theo- 
rem III. 
A ”o 
n + m 
-Mn 
. m + 1 _ 
AO +71 
( 71 - 
I n — m\ A , m- J-2 , q 
. . A 2 0 + &C. 
\m — 2 ; _ _ — 
( to m terms and m factors in each term ) the upper signs tak- 
ing place when m is odd, and the under when even. 
This corollary furnishes an important theorem, greatly 
facilitating the computation of differences. It often affords a 
much more convenient general term for the numerical coeffi- 
cients in the expression for A” u than that given in Theor. III. 
viz. when m is small compared with n in which case the com- 
mon method for the computation of A” o n + m woll }^ be of little 
use. As in the following example : 
