8s 
Mr. Knight on the Expansion 
%dly. Neglect all terms in B which contain any of the c’s 
m+i, 72— -i 
or e’s except c and e ; and proceed, with the remaining terms , in 
the same manner with respect to the d’s. And so on according to 
the number of multinomials . 
The sum of the terms, thus obtained, will give B . 
m,n 
It is scarcely necessary to observe that, when we have got 
a few of the higher terms, by this rule, the preceding ones 
may be found from the equation 
as in the case of one double multinomial. 
To find any coefficient, without a knowledge of the rest, 
when the function contains more than one double multinomial 
we must combine the rule in the last article, with what was 
shewn in article 14.* 
22. Thus we have a complete and simple theory of the ex- 
pansion of functions of double multinomials ; and from equa- 
tions («) and (a) a precisely similar theory may be derived 
for multinomials of higher kinds. 
But it is wholly unnecessary to enter into further details ; 
we are able, without any more trouble, to see what must be 
the solution of the following 
General Problem. 
It is required , in the expansion of any function of a multinomial 
of any kmd, to find B the coefficient of x m y n z r 11 v &c. 
m,n,r,s, t, &c 
from B that of x m+1 y n ~ l d u s v l &c. 
m+ 1, n—i, r, s, t, &c. 
* See Note II. 
