•\-n-\-m . 
of the Binomial Theorem , 319 
n+m - 1 n+m-z n+m- 3 ^ &c which expresses 
234 
the product of the two series into one another. 
17. If therefore r be any positive whole number we can 
raise the binomial series i -f- — x -f- — . 
— . 1 
■^+T’ 
■—2 
x 3 + 
1 1 1 
— - 1 — ~ 2 “ “3 
x 4 + & c. to any proposed 
3 • r 2 3 4 
power by successive multiplications ; or we can express any 
power of it by supposing the multiplications actually to have 
been gone through. Thus, calling the last mentioned series 
the root, if it be multiplied by itself, and if the coefficients in 
the product be expressed in the regular binomial form, its 
— 1 
— 1 
square will be 1+ — x-\- — . — 
2 2 2 
i 2 3 
2 r r r 
r - z - 3 • 4 ‘ ^ 4 + & c - Again, if this series be mul- 
tiplied by the root, and the coefficients in the product be ex- 
pressed in the regular binomial form, the cube of the root 
will be 1 -f jr x+ ~ . r 
— 1 
A-i 
3 r 
— 1 
■* S +T-- 
_ 3 _ J 
r 2 r 3 
3 ^ x 4 -\- &c. Proceeding thus, by multiplying the 
last found power by the root, in order to find the next higher 
r 
— i 
power, the nth power of 1 -f- — x-\- — . 
— 2 
-* 3 + T • 
— 1 
1 1 
_ _2 — _3 
4 
vr 4 + &c. is 1 -f — x + — . 
n n 
r , . « r 
— - — . — 
2 8 r 2 
&C. 
— 2 
M ft 
3 , n r r 
x + T • “I- • -7- 
-3 
<r 4 -f 
MDCCCVI, 
