of any Functions of Multinomials. 63 
The succeeding terms are found with equal ease* I omit to find 
them only on account of the length of the calculation. 
9 . I shall now show that the same method may be success- 
fully employed in more complicated cases; and, instead of 
dwelling on particular problems, shall proceed at once to the 
expansion of any function of any functions of simple multino- 
mials , 
<p ^ F (c -j- c x c x t& ^~ ) if [e e x -j— e x' 1 ), &c. ^ ^a), 
/ t 
If we consider c c x e -j- e x &c. as binomials c -f- y, 
e -j- Zy &c. the function («), which may be, for the moment, 
represented by <p, will have for its fluxion, (y, %, &c. being 
made to vary) 
( j)y + (t) z + &c. = (y)y + (y) z -f &c. ; and conse- 
quent \yj{ (-f-)y-f- ( 7 ) z + &c. } = <p=<p{ F(c+cx + cx'+ ), 
f (e -f* e x 4 - e x* + ), &c. j .... (ft). 
If then we represent the expansion of the function (a) by the 
series 
B 13 .r B x' 4 - B x s + B x n -\- ...... (y) 
123 n 
and denote the fluxional coefficients, of the first order of 
c c c 
B, B, B, &c. with respect to c thus B, B, B, &c. ; 
12 12 
e e e 
with respect to e thus B, B, B, &c. ; 
1 2 
& c. &c. 
the equation marked (/3) will become 
F ip 4" c x ~\ ~)>f { e "1“ e x +), &c. | —f\ B"fi B x + B x *-\- 1 
