of a Series in the inverse Method of finite Differences . 123 
71=1, sufficiently shew of what importance the enquiry has 
been considered. 
3 m 
Theorem. ux— — -\-a ^h^bfh^^...V-fh m -^SiC 
O 2 X X X 
in which the even powers of h are not found, and P the coeffi 
m 
m— 3 
1 / nz— i , m ^m—z m . 
01 =h m = .■ — a A 0 —2 A +< 
=— 1)2 ^ 
.3 m , . m m\ 
A 0 ....-{-A 0 . 
m 
Demonstration. The coefficient of ~ h m , or f Yl+l ' } ( Vid . 
x m 7 v 
Theorem V .) is the coefficient of h m+l in the expansion of 
f 1 + Zi+7X- 3 + &c -) _, - or of (4 
1. Now 
JL- h - 
T ,~h 
(t) 
Let 
represent the ex- 
i+A/i+BA* +NA" , +P/i m+, + &c.< panS1 °" of «~’ 
being anyoddnum- 
Lber. 
^ en Twill represent the 
— 1 -f Ah — m\ . ., . +N h m - P h m + 1 + &c.<! 
'W+I 
* 
i expansion of — 
Hence by equation ( 1 ) we obtain 
A+C/i\.4N/t m “ , + &c. = -{. . . . (2) 
From whence it follows that 
A=f, C=0.. s .N=0 . . . . Hence when m is any odd num- 
ber the coefficient of h m = 0 , and therefore the coefficient of 
m — 1 
. 771—1 
.771— I 
X 
MDCCCVII. 
S 
