72 Mr. Knight on the Expansion 
If the function under consideration contain three multino- 
mials, the origins of derivation will be 
'r V — i ' 'r— 2 '2 '2 V — 2 
c , c e, c e , .... c e 
j r V— i V— 2 ' V— 3 '2 ' 'r— 2 
a x (c , r e, c J <? , r £ 
j2 / ir— 2 ' r — 3 ' 'r— 4 '2 V- 2 \ 
n x (c , c ^ r, c ^ , e ) 
/ / _ / / /_ 
X (r , c e , e ) 
x (c , ^) 
d r 
and all the possible combinations derived from each particular 
origin as,r w <4 will be expressed by the product 
(c n + A c n A 2 c n -f, &c. ) 
(> + A^ + A 2 i ? 4&c.) 
The reader will easily extend the method, if necessarjr, to a 
greater number of multinomials. 
As we have, in this manner, a certain, easy, and regular way 
of finding all the combinations in any coefficient B, the pro- 
r 
blem is completely solved. 
I go on to multinomials of higher kinds : and, with M. 
Arbogast, shall call those multinomials of the nth order which 
are disposed according to the powers and products of n differ- 
ent letters x, y, z, &c. 
15. After having so fully entered into particulars, in the 
preceding cases, there can be no difficulty in perceiving, that 
a complete theory of derivation, for the expansion of any function 
of any number of functions of multinomials , whether they be simple , 
+ 
A e m 4 A 
2 e m 
&c. } 
