of any Functions of Multinomials. 
% 
/ li ° 
all those which contain c or c\ and so on. In B must be neg- 
n — i 
e 
lected all terms which contain any of the c’s ; in B these, and 
n — 2 
those also containing c ; and, in general , all those terms must he 
neglected , as we proceed, which contain any quantities whose flux- 
ions have entered into the preceding terms. 
From the above equation is derived an easy mode of expan- 
sion, I shall give an example in the case of two functions ; and 
shall represent, for brevity, <p |f ( c),f (^)j by®, and its 
fluxional coefficient of the m «th order (when the fluxion 
has been taken m times with respect to c and n times witli 
m,n 
respect to e) by <p . 
1,0 , 0,1 , 
We find here, B = <p ; B = <p c -f- cp e ; 
i 
1,0 „ 2,0 g 0,1 „ 0,2 g I, I , , 
B = q> c -J- cp — -j- (p e -j- cp — -j- <? c e ; but to explain more 
2 
fully the manner of proceeding, let it be required to find B 
3 
from the preceding coefficients. We have, after neglecting 
the specified terms. 
/ lC i 2,0 f a , 1,1 f u 1,2 , t 2,1 t t s-frC n 1,1 i, r 
B c = (p c c ^ cp f <p c e <p c f q> fe ; / B c-— (pee 
2 2.3 2 2 e/ I 
/ iC m 1,0 lu , o, 2 , (( 0,3 y s~tiC w 0,1 m 
B c m =s q> c; J B e-=. (p e e -J- <p — J B e- = <p e ; the sum of 
these gives 
1,0 
2,0 , „ 3,0 
0,1 
c 3 * III °’ 2 1 II °>3 p 1,1 I 11 11 1 
B = (p c (P c c ^ (P ~ 4- <p e ^ cp e e ^ <p — + cp[ce ce) 
3 
1,2 ' p 2,1 ! 
+ QC- + Q -e. 
MDCCCXI. 
K 
