of a Series in the inverse Method of finite Differences. 1 39 
Theorems relative to the inverse Method of finite Differences . 
Theorem IV. Let u be a function of x and x-\-h, x-\-zh, &c. 
successive values of x, as before, then 
S n u—(e * — 1) if after expansion 4 - ) , 4 - &c. are 
changed into fl." ux n , ax n ~\ See. and | 4 -j , J-jj , &c. 
3 
into - 4 , 4 ?, &c. 
x x J 
Demonstration. By Theorem II. 
? M+I n+i H+Z n + z 
( 1 ) — u — A (S» if h n -\~B ( S n u) h * 4 *C ( S* u ) h -J* Scc„ 
^ x n+I X n + 2 
where A, B, C, &c. are the coefficients of 1, h, h 2 ', See. in the 
expansion of (-y- ] — ( 1 "r 7^ + ”3 ~K & c - )”• S° that if v 
. 2 
&c.)”, making li— o, v=A, v=Bh, vz=C li 
Sc c. 
n 23 
(a>--Let*(^-fe««+j 0 i : A+yiA t +^A , + &c. It 
x n 
is evident that this assumption may be made to satisfy the 
equation (1). 
Then, by taking the successive fluxions of equation (>2) 
and substituting in equation ( 1 ) , we have 
* Euler uses a similar assumption in his investigation of the sum of a series 
from its general term, p. 15, Tom. VIII. Com. Petropol. 
