of the Binomial Theorem. 
321 
n 4 - m — 1 
n-\-m — 2 3 
— X 
__ |~ ' * a j tl -4- Wt— I n - ' 
— 1 — x . 
. — - — - . — ^ — . — — — - x 4 — &c. 
Hence it is evident that if the series equal to 1 — x\ n be 
multiplied by the series equal to i—x\ m , the product must be 
equal to the series, which is equal to 1 — x\ ,l+m . Now the two 
first mentioned series being multiplied into one another, and 
the parts being arranged according to the powers of x, the 
several products will stand as in the following representation. 
= 1 — n x- 4 -n . z° 
* 2 
n , m — T 
* 2 
n — 1 ft — 2 , , 
n . . x + n . 
2 1 > 
„ m — 1 m — 2 3 , 
x* — m . . x -m. 
2 
3 
. X 
4 
m — 1 
m — 2 
m ~3 4 
2 
3 
• *A - 
4 
Tl 1 
n — 2 
* 
b 
1 
2 
3 
• *>0 
4 
n — 
n. 
1 
n — 2 4 
— X 4 — 
See. 
See. 
+ n — 1 „ n — 1 n — 2 3 , 
U . X * — n . . X + 7 Z . 
2 2 2 ■ 
— mx-\-m . nx 1 — m . n . 
2 3 
n — i 
m — i 
m . x — m . 
2 
m — i 
x 3 -\ -m.n 
3 I m — 1 
. nx -\~m . — — . n 
3 
n — i 
m — i m — 2 3 , 
-m . — — . : X . 
2 2 
m — i m — 2 
■ . nx 
3 2 3 
For the sake of reference hereafter let this be called 
multiplication B. 
Now for the same reasons as are stated in the 15th and 
ifith articles, the whole coefficient prefixed to any power of 
x in multiplication B, must be equal to the coefficient prefixed 
to the same power of x in the series 1 — . 
&c. 
&c. 
&c. 
&c. 
m\n — 1 2 
x — m-\-n 
vi-\-n — i m-\-n — 2 3 
x -j . 
m-\-n — 1 m-f 72 — 2 
™ + n — 3 4 
x * — &c. ; and we are also at liberty to suppose n and 
m to be any two fractions whatever, in the series multiplied 
into one another, and consequently in the series expressing 
their product. 
T t 2 
