of a Series in the inverse Method of finite Differences. 12, & 
Then we shall find that 
(>) 
n (1) 
— - m = m. a 
m a 
m{m — 1 ) ( 2 1 
-J- — l 
1.2 
, ^(3) , 
' 12 3 ' ’ ‘ * 
(m— 1) . (m) 
ma K ~f” a 
For taking m quantities a, (3, y, Sec. we easily deduce by help 
of the multinomial theorem, 
(«-H3+y+&C . )’■ = + 0- + &c + JJ 0 + &c . + 
f of" y &C. -{- &C. 
where p-\-q=n, p-\-q-\-r—n, Sec. &c. 
Now if we consider a term j J fff Sec. where k quantities a, 
(3, y. Sic. are concerned, and if p' ,p' . . ( t' numbers), q' , q' . . (y f 
numbers), r' , r' . . (w' numbers), denote values of p, q, r, Sec. 
satisfying the equation p-{-q-\-r-\- Sec. ( k terms) —n, we shall 
see, that the number of the terms, in which these values 
p' ,p', . . q ' , q . . r', r' , Sec. are found, is the number of combi- 
nation of m things, taking k together into the number of per- 
mutations of k things, of which t' , v', See. are the same. This 
is evident, because in any product ct(3y [k factors), the indices 
P',p',..q',q, Sec. are to be annexed in every possible order. 
And when the quantities a, 13, y, Sec. are each units, each of 
the quantities is expressed by the same quantity, i 1 ' . i pl . . 
. . V . i’ 7 &c. therefore the sum of all of them is 
m(tn — i ) . . . Im — A -}- i ) 
Hence, 
m(m— i) .... (nz— £-J-i) k ) 
" 7 ™“— — " 1 CL 
1 . 2 . .k 
1.2 . . k 
when u, i Q, y, Sec. are each unity, 
/ cf 0 1 y &c. -f” Sec,, [k quantities) 
Hence by substituting for k the numbers l, 2, 3, Sec. we 
easily obtain the equation (1). From which, substituting for 
m successively 1, 2, 3, Sec. we obtain 
R 2 
