of any Functions of Multinomials. 8 1 
fluxion with respect to all the quantities hut ; change generally 
into Q f + j and take the fluent with respect to this last. The same 
terms must be kept only once. 
By this rule, we find B beginning with <p (c) x * *»°J x 
m,n 2,3 ...m z.^...n 
for origin of derivation : the reader may compare it with that 
given by M. Arbogast at p. 113 of his work. 
21 . If the function to be expanded contains functions of many 
double multinomials, all the formulas, and rules, that have 
been given for one, may be extended to this case, by means 
of equations («) and (a); in the same manner as alike ex- 
tension was made in treating of simple multinomials. 
Thus, from the method of finding B given in article 19, we 
m,n 
get the following 
Rule. 
To find Bfrom B in the expansion of any function of any 
m,n m J r i,n — t 
functions of the double multinomials 
c 4- i!o x + ’ e + i%* + ; ^ + \,o x 4 - 9 &c. 
+ o^y + + 0% y + + 0,1 y + 
+ + + 
1st. Consider only the c’s, and take the fluxion of B with 
l > w— i 
respect to all of them except c , 0 ^, 0 C 2 , &c.; and proceed exactly in 
the same manner as zuas directed for one double multinomial in 
article 19. 
zdly. Neglect all the terms in B which contain any of the 
i, n— 1 
c’s but c; and proceed, with the remaining terms, in the same 
manner with respect to the e’s. 
MDCCCXI. M 
