of the Binomial Theorem. 
3*9 
- — 2 , — - 3 
Z 3 X r +, &c. being multiplied by the series x r -f~ 
z’x ' 5 
- r zx’ l +y*.L—z'x f +^ x i__ x 
W + 1 " + 1 _ j 
-j-, &c. the product will be x r + r ' + —• x 
»+> 
z X 
+— x ,X-I 
1 r 2 3 
> lxi„ 2 1+1 _ 3 
+ ,&C. 
so. From hence it follows, that x + z} r = x r -j- ~ zx r 
n n n n n 
I ~ — 2 3 
I n ~r 1 3. — — 2 , n 
+ 7 xL T u + T x 
.% 3 x r -j-, &c. For 
as » in the last article may denote any number whatever, the 
square of the series x r -\-~zx T -fy 
* +T 
| , 2 — ■ - _ j , ^ | 
X I — _ X .1 — - z 3 x r 4-, &c. will be x r 4- — zx T + 
23 r 1 
_ 3 
x l z 1 x r 4“ t x — — x 1- , z 3 x r +,&c. and this 
being multiplied by x r + ~ zx r + — x 1 - x r -f 
-3 
z 3 x r +, &c. the product will be x r + 
3 J J , J 2 3 J J 2 3 , 
~ZX r + 7 X Z* X r + -f X I- X — 1 Z 3 X r 
r r 2 r 2 3 
4-, &c. 1 being added to the numerator of the fraction of 
which r is the denominator, upon every multiplication. The 
Tts 
