of the Binomial Theorem . 
3°3 
multiplication will appear as follows, and equations of various 
dimensions will arise, according to the powers of x. 
x + a =p 
X + b _ = l 
X '+ a b }x + ab 
X - c — r 
p q; a quadratic equation. 
x 3 + a 
T “I + a ^1 
-j- b 1 x'-\- ac\x-\-abc = pqr\ a cubic. 
-1- eJ 4 - b c J 
-{- c J b c 
x + d=s 
■z 4 -f a~\ -f a b 
b l ^3+ * c 
+ 
%d\ 
>x 
+ be* 
-f ad | 
+ bd I 
•fair 
i-j- a bd i 
-j- a c d | 
-j- 6 c 
jr -{- a b c d = p q rs; a biqua* 
dratic. 
-f- a, b- 
-j- a c 
-j- c »c 4 -{- b c 
-j- d | -j -ad 
z 5 -f a 
+ b 
+ 
+ bd 
+ cd 
4 - a e 
4 -be 
4 - c e 
4 - d eJ 
>x 
-{- a b c- 
4~ a b d 
4“ a c d 
4 -bed 
j 4~ a b e 
4 -ace 
4~ bee 
4 - ad e 
4 - bd e 
4- c del 
&c. 
-j- a b c 
4- a b c e\ 
a i L j l 00 ~r w u i/ w t — 
\ X ~\_ a j | pqrst; asursolid. 
-f - a c d e\ 
4 - b c d e J 
-j- abode — 
4. From the above it appears, that the coefficient of the 
highest power of x in any equation is 1 ; but the coefficient of 
any other power of x in the same equation consists of a certain 
number of members, each of which contains one, two, three, &c. 
R r 2 
