12 5 
of a Series in the inverse Method of finite Differences. 
that = b {m ~ l) + {m-i) b (m ~ 2) + lH 
b {m ~ 3) + &c. 
&c. &c. 
These equations are the same as the equations between the 
terms of a series of quantities, and the first terms of their 
respective orders of differences ; 
. . . a^ m ~ a^ m \ corresponding to the terms of the series, 
and . . . b^ m ~~ 1 ^’ b^ > to the first terms of the respective orders 
of differences. 
Whence we conclude, that 6 (w) = a ^ — ma ^ m ~ 1 > -f- 
’ 1 1.2 
fm— 2 ) — ^ — (by the preceding lemma) 
. m n ni — l n— I . m(m — i) A m — 2 n — 2 0 , 
= A 0 — m A 0 q — - - A o — &c. and 
_ 1 12 
therefore, excluding unity from the values of p, q, &c. 
(l*)” (r)” &C. = A m o‘_A”'-V , - , + r ^ A m ~ z £~ 2 
. . . . (w terms). 
(0 
Theorem VI. Let S" n — f fl. x" 4 - fl. w 1 u x n 1 
h n 1 .n — i 
P 
+ & c. ( vid . Theor. V. ), then or the coefficient of the m-\- i 
°™ + 3, — &c. (tom terms and m factors in each term ). 
Demonstration. (By Theorem V. ) The coefficient of the 
