of any Functions of Multinomials. 
6f 
\..m 
c, c, c, &c. e, e, e, &c. &c. and change , every where , c • into 
"...(ra-fi) , "...m "...(m+i) 
c x — — , and e ’intoe x 
» n 
The reader may try this rule on the examples in the last 
article : I proceed to simpler methods for practice. 
By combining equations (0) and (e), and considering B 
(Oi ( 2 B (3), 
under the forms B = /3 -f- /3 -f- /3 c*-\- /3 d- {-, and B = A + 
n — 1 m— 1 
(0 , (0 , ( 3 ) , n c > Oi 
n; » t 2 ; , u; , 
A * + A «* + A e* + , an( i & c - we find B = / B c* +/ /3 r-f- 
n h_i 
111 /^ ( 3 ) II II F e ' /’•(Off III 
£/ {iC’+QSf / 3 C' + + / Br-f / A <?•-{- 2/ Ar-j-2.3 
c cc h__ x e 
/ ($) 1111 (0 
A** + + &c. where in /3 all terms must be neglected 
which contain c ; and in general, according the rule given in 
article 10 , all terms must be neglected that contain any quan- 
tities whose fluxions have entered into the preceding terms. 
By this method the expansion might be accomplished with- 
out difficulty, each term is found at once, and no reductions 
are necessary : the one which I am going to give is, however, 
much better, being, I conceive, the simplest possible. 
12 . By combining the equations (^) and (e), there results 
