go 
Mr. Knight on the Expansion 
- =f( c ) c +f ( c ) 
/ hi 
c c 
III r Z II III / 
+/(04-*+/(0 
2.3 4 
• 2 
&c. &c. 
8. To complete the theory of the expansion of any function 
of a simple multinomial, there remains, for us to solve, the 
following 
Problem. 
It is required to find B without knowing any of the coefficients 
n 
that precede or follow it. 
It is, in the first place, evident enough, from what has been 
done, that 
"•••» // n w w 1 ) 
^ — / ( c ) c +/ (0 ^ M ^ + + J (0 ' 4 ' 
n 
"...m "... m "...w '« 
+/ (0 ^ + +/M “3.4.... 
"...m 
where consists (without considering the denominators) of 
/ 11 111 
all the combinations that can be formed of c, c, c, &c. in which 
the sum of the strokes shall be n* and the sum of the expo- 
nents m. But to form these combinations, for the higher 
powers, would not be very easy. It may not be amiss to in- 
quire, therefore, for some regular method of immediately 
— 1) "...m 
deriving vk from ; so that we may get all the \|/s 
, "...M 
‘n 
successively, beginning with ~~~; l which multiplies/ (r). 
1 # 111 
# I mean when the powers are expanded, as when c 3 is written c c c. 
