So Mr. Knight on the Expansion 
Problem. 
20. To find B immediately , without knowing any of the other 
m,n 
coefficients. 
The coefficient of x m y n will easily appear, from what has 
been shewn, to have the following form; 
£ = (p (c) fi n + <p ( c ) 4 ' + .....+ <P (0 4 + (c) 4 +-• 
m,n 
/ c \m re \;i "...r 
4 * (0 x iiisL x i2iiL where contains all the combina- 
2 . 3-., 272 2.3 .,.22 
lions that can be formed of the r’s (after c or 0 ^ o ) in which 
the sum of the bottom figures, on the left of the commas, is 
rn ; the sum of those on the right n : and the number of fac- 
tors r. Moreover every power as the mth. will be divided by 
2.3 ...m. 
And the reader, who considers how the similar problem 
was solved, in the case of a simple multinomial, will have no 
difficulty in perceiving the reason of the following very simple 
/ 
Rule. 
"...(r— 1) "...r 
To find A from \p , 1st. take the fiuxional coefficient, of the 
latter , with respect to and , of this fiuxional coefficient, take 
the fluxion with respect to all the quantities; change generally 
f y into r and take the fluent with respect to this last = 
"...r l c f ( c W ( c Yx( e \ v 
%dly. Anv terms in \b of the form \ no/ x\2d' ' \ o,u) x &c. 
2'3-p 2 y -q 2.3 .. ,t 
where , except in ,* 0 , all the figures are on the right of the commas, 
will require, besides , the following process. Take the fiuxional co~ 
efficient with respect to f, and , of this fiuxional coefficient , take the 
