of the Binomial Theorem, 
323 
tn~ H— i . , m — n—i in—n—z , , — ■ 
* x*A-m—n . — . x A- m — n . 
2 1 2 3 * 
m—n — 3 4 
m — «— 1 wt— n — 2 
&c. ; and as this equation holds in every pos- 
sible value of m , and as, by the general principles of involution 
l-f-rl* is equal to 1, when m is equal to 0 then or 
i+*i 
n =i —nx—n. 
r*=2_.=!!=l a ,*-- & c. 
3 4 
According to the form of the binomial series, the whole of 
the second, fourth, sixth, &c. terms in the last series consist 
of an odd number of negative parts multiplied into one 
another, and therefore each of these terms becomes a nega- 
tive quantity. But the whole of the third, fifth, seventh, &c. 
terms, consist of an even number of negative parts multiplied 
into one another, and therefore each of these terms becomes 
a positive quantity. Consequently, i-\-x\~ n =zi—nx-\-?i . 
— n— 1 
■n . 
I — 11 — 2 
-n . 
-n— 1 
n + 1 n-\- 2 , . n-fi « + 2 11 4-3 . 
-n . . - 1 — x*A-n . — L ~ . ^ x - 
2 3 1 2 3 4 
&c. 
si. By the 19th article we are enabled to prove that 
3 , m — 1 in — 2 m— 3 
4 -f-m . — . . — - 
jn § 1 772 ■ 1 1 . 1 
1 — X \ i-j -m . — x-\-m . — — 
m — 1 tw — 2 
2 * 2 
-or- 
1 — xf i -| - n . — x-\ -n . 
n—i . n — 1 
a fl - 4 - n . ■ — - 
2 & 2 
71 2 
3 
■* 2 3 -f n 
n ~ z ”-3 
3 ’ 4 
is equal to the series l-f -m—n . -—.r-f-m— n . 
r?7 — 7 / ~ 1 
— n 
m—n—i m — n — 2 
— -x -\~m — n . 
m — n— 1 m — n — 2 in — n — 3 
2 3 * 2 3 
&c. For, as in the preceding article, if this last series be mul 
tiplied by i-j-w . -x-\~n 
n— 1 „ , n — t 11 — 2 
x -f- n • • — , 
2*23 
■x~ 
71—1 
7 ~~ . 4 -j- &c. the series expressing the product will bs: 
i-J -m x-\-m . 
m- 
x 1 -}- m . 
171 - I 
2 
3 1 in — i 
x A- m . 
• z 
&c. 
a ,4 4 & c . 
