of any Functions oj Multinomials . 73 
or double , or triple , &c. is contained in the following equa- 
tions. 
/» c c p e 
B = £./ B xpt,j/,p,6cc„-f« E. f B 
m, n, r. See. m—p, n—v, Sec. m-p, n~v, r—p,8cc. 
X pt, v 5 p, &C. &C. ....... (T) 
B 
B 
| m,n,r,8cc. 
&C. 
(x). 
m—-u.,n—»,r—p, See. \ . / 
r I'iP* &c> / c 
where B is the coefficient of &c. in the expan- 
m, n, r, Sc c. 
sion ; ^ &c. and pi & c. &c. are the coefficients of x* y z p 
&c. before expansion, under the signs of the functions. 
The sign X, in the first formula, expresses the sum of all 
the terms that can be formed by taking for p, v s p, &c. all the 
whole numbers from 0 to m, n, r, & c. v, p, &c. must not how- 
ever all equal nothing at the same time. 
It is scarcely necessary to observe, that certain terms are 
understood to be neglected in equation (»), according to the 
rule given in article 10, which is, that all terms in the B's must 
be neglected, as we proceed, which contain quantities whose 
fluxions enter into the preceding terms. 
The above expressions if considered not only in themselves, 
but with respect to the formulas that are immediately dedu- 
cible from their development and combination, in the manner 
that will presently be shewn, appear to be the most general 
and important in this branch of analysis. 
16. Let it be required to expand the double multinomial 
function 
MDCCCXI. 
L 
