§7 
of any Functions of Multinomials. 
Suppose any one of these origins to be 
(A). 
2.3 ..r 2.3...S 2.3 ...t 2.3...U 
Let A) A% A 3 , See. represent the successive derivations made according to the rule 
in article 20. It is plain that all the terms got from the origin of derivation (A) will 
be expressed by the product- 
multiplied by 
In this manner may the terms be derived from all the origins ; after which we have 
only to arrange them under their appropriate fluxional coefficients. 
If we wanted to find immediately B in a function of two multinomials of a 
m, n, r, Sc c. 
still higher kind, the method would be exactly similar. 
Note III. In the preceding pages, I have considered the expansion of multinomial 
functions generally ; and abstained from giving particular examples, that the paper 
might not be extended to an unreasonable length. There are, however, some cases, 
— when the function is a whole positive power -— which require a separate notice. The 
method of direct derivation given in article 5, and a similar one at the end of article 
11 will here fail : this indeed is of no consequence, as the rules in article 6 and 12 
are both easier than the former, and applicable to every case. But it will be necessary 
to give new methods of inverse derivation 1 for if we consider those in the paper, in 
article 7 for example, it will easily appear, that though they are true generally for the 
rath power, the case is very different when we give to this letter the particular values 
1, 2, 3, See. The reason of which is that the fluxional coefficients off (c), after the 
first, or the second, or the third. See. vanish ; and these functions may be said not im- 
properly, when compared with the general form, to give defective expansions ; any 
rules, therefore, which depend on the depression of the fluxional coefficients of f (c) 
will be of no use here. 
The following very extensive rule is the reverse of that, for direct derivation, in 
article 12. It agrees, in its simplest case, with that of M. Arbogast in his article 36. 
