of the Binomial Theorem. 315 
their repetitions, in it. But as the number of quantities in 
each member of a coefficient is 1 less than the number in each 
member of the coefficient next following, each member of 
the coefficient whose number of members we are now calcu- 
lating will have in \tp-\~\ number of quantities. Consequently 
- u ~ psn = x = the number of members of the coeffi- 
pXp-rl P P + 1 
dent next after that whose number of members is •?, as in 
P 
the last article. 
12. It is evident, from the sixth article, that the value of 
p in the second term of any equation is 1 ; in the third term 
of any equation its value is 2 ; in the fourth term of any equa- 
tion it is 3, &c. It is also evident that the number of members 
of the coefficient of the second term of any equation is n ; for 
the whole coefficient is the sum of all the quantities a, b, c, &c. 
used in producing the equation. It therefore follows that the 
general expression ■— x obtained in the last article, enables 
us to ascertain the number of members in the coefficient of 
any term in an equation. For the number of members of the 
coefficient in the second term being n, according to the suc- 
cessive values of p the number of members in the third term 
is n . ^ ; in the fourth term it is n . ; in the fifth 
2 23’ 
• • 71 — — I 71 2 7 \ —— 2 . 
term it is n . — ; and this regular form may be 
extended to express the number of members in the coefficient 
of any term whatever. 
13. The binomial theorem, as far as it relates to the raising 
of integral powers, easily follows from the foregoing articles. 
For if all the quantities a, b, c, &c. used in the multiplication, 
in the fifth article, be equal to one another, and consequently 
