of any Functions of Multinomials, 
% 
li 2 i2 y gi 2 ,J I II I 3,1 Jj / / /// *’2 1 1 II l >3 ; g 3 
4 " ^ c "7 + ^ TT " 5 ” ^ + <P — <?>£—. 
1 he second part gives 
°>3 > " 
°’4 
e-f (p . 
■ ' ?.. 7. A 
°5 I HI I °J 2 / HI g Z 
»'+f(»+T)+», , . 2 . 3f - 
The sum of these is B. As the number of multinomials adds 
4 
nothing to the difficulty of expansion, according to this me- 
thod, it is useless to give more examples. 
13, Nor does the number of multinomials make any diffe- 
rence as to the facility of inverse derivation ; which depends on 
the equation 
n—m 
Thus from B, just now given, in the case of two multinomials, 
4 
let it be required to find B ; we have 
no 
- „ 2,0 y 0,1 „ 0,2 1, 1 , , 
cpc^tp — -j-cpe°j~(p---j r <pce« 
4 - 2 \ ;./ I 
14. There remains the important 
Problem. 
To find B without knowing any of the other coefficients. 
r 
It will be plain to any one, who in the least considers the 
methods that have been employed, that B must contain all the 
I II III I U HI I H III 
possible combinations of c, c , c, See. e, e, e, &c. d, d, d, Sc c. See. 
that can be formed with this condition, that the number of 
strokes be r. Every mth power will be divided by the product 
