C 49 J 
On the Expansion of any Functions of Multinomials. By 
Thomas Knight, Esq. Communicated by Humphry Davy, 
Esq. LL.D. Sec. R. S. 
Read June 7th, 1810. 
1. 1 he expansion of multinomial functions has, of late, been 
so ably and fully treated by M. Arbogast, in his learned 
work ‘ Du Calcul des Derivations’ that it may appear, perhaps, 
scarcely necessary to add any thing to what has been written, 
on the subject, by that excellent geometer. 
Nevertheless, as he is the only one that has hitherto culti- 
vated this part of analysis with any great success ; and as it is 
agreeable, I believe, to most persons, to be presented with 
various solutions to mathematical problems, I hope it will not 
be thought superfluous if I show how the same things may 
be accomplished in a very different manner. 
By the procedure here made use of, we shall also be ena- 
bled to arrive at many new and remarkable theorems ( both 
for direct and inverse derivation) , which could not, I imagine, 
be very easily found by M. Arbogast’s methods. 
For a function of one simple multinomial, I give (amongst 
others) the same rules of direct derivation, as that author ; but 
when there are many, and in the more difficult cases of double 
and triple multinomials, &c. or functions of any number of 
these, I offer new and expeditious methods ; which are de- 
monstrated with the less trouble, from the analogy which 
MDCCCXI. H 
