of any Functions of Multinomials. 
71 
we are seeking) derived from the origin c n e m 3 will be ex- 
pressed by the product 
(<?*+ A c H + A 2 c n + A 3 c n + ) ( e m + A i m + A 2 e m + A 3 e m + ), 
where each derivation of e m is multiplied by every one of c n , 
and conversely each one of c 11 by every one of e m . 
We have nothing to do then but to deduce the derivations 
from c n and e m by the rule in Art. 8. 
Suppose that B was the coefficient required, and that we 
5 
wanted all the combinations arising from 
1 . 1 _ 
c 3 e . 
We have here 
'2 2 2 
A c 3 = c C ; A c 3 
% % 
c; A 0 c 3 ~~ 
. '2 . 2 2 
= 0; A e — e ; A r =o, 
and, by substituting these values in the above product, we find 
all the combinations arising from the origin c 3 e 2 to be c 3 e 2 -f- 
/ // ; // / /// / 1 11 11 w ii 
c 3 e -f- c c e -\- c e -\-cce-\-ce. 
It would be as well to write down the appropriate denomi- 
nators to each combination as W'e proceed : and when we had 
treated all the origins of derivation in this manner, there would 
only remain to arrange the terms under their proper fluxional 
coefficients.* 
/ , . n,o, om, 
* Instead of c n e m > I might have taken for origin of derivation <p c n x <p e m > and 
after multiplying the factors 
n,o , n—1,0 , n—2,o , o,m , o,m-—i , o,m— 2 , 
(<pc n + <p .ac s + . a 2 c w +) ( <p e m -i- <p .Ae m + q> ,& 2 e m +) 
1 >,0 0, |!4 V^Jt, 
have changed <p x into <p ; but this would only give additional trouble without 
answering any useful end ; it is sufficiently plain that the appropriate fluxional coeffi* 
cient of A' c n x t m will be 9 
