3H 
Mr. Robertson's new Demonstration 
m — 2 m — 3 4 
tf 4 + &e. Consequently as ~ y = i -f- w — 
— — — . m — n — j 
— -.r-j-m — n .~ — a* -j- m — n . 
I— x\ 
m — n — i m—n — z 
. — — % 
m — n — i- m — n — z m — n — 3 
&c. in every possible value of 
m, it follows that when m is equal to o, then -- - --- or 1— 
l— x[ n 
— n 
— n — 1 „ — n— 1 —n — 2 j —n — 1 
■ n . u 2 — n . . . — x — n . 
■ — 71 — 2 — n — 3 4 0 
J X &C. 
3 4 
The form of this series, however, may be changed into one 
more convenient. For the whole of the second, fourth, sixth, 
&c. terms consist of an even number of negative parts multi- 
plied into one another, and therefore each of these terms 
becomes a positive quantity. And as the coefficients of the 
third, fifth, seventh, &c. terms consist of an even number of 
negative parts multiplied into one another, and as in these 
terms the powers of x are positive, each of these terms be- 
comes a positive quantity. Consequently i- — x\~'‘ n =i-\-nx-\- 
n 
M+ X 
x 2 -{-n . 
71 4 - I 71 + 2 
jt 3 -| -n 
71 - f I 71 4- 2 71 + 3 
x 4 -\- &c. 
Every particular necessary for the establishment of the 
binomial theorem has now been proved. I therefore proceed 
to conclude the subject, by shewing that each of the four 
forms, in which the theorem may be expressed, immediately 
follows from the preceding articles, and the general principles 
of involution. In each of them n is to be considered either 
as a whole number or fraction. 
22. By article 18, i-j-xr=i -\-nx-\-n 
4* n 
-2 n — 3 „ . „ ™ b 
71 — I z , 71 — I 71 — 2 , 
X + 77 . . X 3 
* 1 2 3 
a 4 -j- &c. But if — be equal to x, then 
3 
