go8 Mr. Robertson's Demonstration 
ber of members in the coefficient, when exhibited in the man- 
ner of the 3d article. And as n expressed the number of 
quantities a, b, c, &c. used in the multiplication, when each of 
these quantities is equal to a, it will denote the power of the 
binomial x a. 
Hence, if m denote the numeral coefficient of any term of 
the rath power of x -j- a, and p the exponent of a in that term, 
the numeral coefficient of the next term will be expressed by 
m x as is evident from the last article. 
11. It is manifest from the 3d article that x -f a being 
raised to the nth. power, the series, without the numeral coef- 
ficients, will be x n -f cl x' — 1 -f- a 1 x r ‘~~ 2 -\- a 3 x n ~ 3 -J-, &c. and as 
the coefficient of the first term is 1, and of the second n , from 
the general expression in the last article x + d n = x n -f 
?iax n 1 4 - n x a x n ~ 2 4 - n x x a x”—* 4 -, &c. 
I 2 1 2 3 1 
12. If equations be generated from x — a x x — b x x — c 
xx — d, &c. the coefficients will be the same, excepting the 
signs, as those which result from x-{-axx-{-bxx-\-cx 
x -f d, &c. in the 3d article ; and as — x — gives -f > b ut 
■ — x — x — gives — , the coefficients, in equations generated 
from x — ax x — b x x — c x x — d, &c. whose members 
have each an even number of quantities will have the sign 4> 
but coefficients whose members have each an odd number of 
quantities will have the sign — . And hence it is evident that 
x — = x — n ax 1 -j- n x—~ a x n ~~ 2 — n x -j- x — — 
c f x n ~ z +, &c. 
13. Having thus investigated the binomial theorem, as far 
