of a Series in the inverse Method of finite Differences, 231 
m 
efficient of — h m in the expression for S u = 
m 1 
( m A - 1 \ m 4 - 1 
I.2.3. m \ 2 — 1/2 
2 
AO 
S . 2 W « j . ?H ? 
A 0 -j-A 0 
The same conclusion may be derived somewhat more 
easily by the assistance of a diverging series, as follows. 
This investigation, however, is not given as affording the 
same satisfaction to the mind as the above demonstration, 
-i—e e — &c. 
h 
e +1 
Hence it is easy to see that the coefficient of h m 
= — i m 2 m — + &c. in infinitum. 
Also _i_ = -flL = e - b -e- Ib J t . e -l b - &c. 
6 ~/t» * 
e -J-i i-fg 
from which it likewise appears that the coefficient of h m 
= ± 1* + 2 OT ± 3’" T Sec., the upper signs taking place when ni- 
ls even, and the lower when odd. Hence when m is even 
— ] W ~J-2 W — 3“+ Sec. =i m — 2“-|"3 ?h — Sec. and therefore neces- 
sarily — i m 4- 2” — 3 W -f- Sec. =0. Consequently, when m is 
even the coefficient of h m in the expansion of ~ — ■ — o. And 
e° -f-i 
generally by the appplication of a well-known theorem, 
m . in m , 0 i i m , 12m 0 
— 1 -f- 2 — 3 + Sec. = - A off- — Ao ■— &c. 
Whence, &c. &c. 
It may be remarked, that in the above theorem the coeffi- 
cient of h m c an be computed, without using a higher quantity in 
the series o”, i m , 2 m , &c. than For the first terms of 
the^i + 1) ^ + 2) &c. orders of differences of the series 
I m — im I l m — i\ \m I Im — 1 \ m 0 . 
(—7-) , (i — {— j) , (2 — [— jj , &c. are obtained 
