of the Binomial Theorem. 307 
whose number of members we are now calculating will have 
i n it p -j- 1 number of quantities. Consequently ~~ t>sn = 
~ x Lnii— the number of members of the coefficient next after 
that whose number of members is y, as in the last article. 
The same conclusion may be obtained in the following man- 
ner. Let m = the number of members in a coefficient,^) = 
the number of quantities in each member, and n — the num- 
ber of quantities a, b, c, &c. Then will mp express the number 
of quantities with their repetitions in this coefficient, and 
the number of times each quantity is found in it. Hence, as 
each quantity is only found once in the same member, m — • 
= the number of times each is not found in this coeffi- 
n 
cient, and is therefore equal to the number of times each is 
found hr the next coefficient, according to the 6th article. 
The number of quantities, therefore, with their repetitions, in 
the next coefficient is expressed by m — y.n = mn — mp\ 
and as the number of quantities in each of its members is de- 
noted by p -j- 1, the number of its members is expressed by 
mn — mb n — b 
" — m x f+ 7* 
10. 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 t &c. used in the multiplication 
in the 3d article, be equal to one another, and consequently 
each equal to a, each of the members in any coefficient will 
become a power of a ; and each term in an equation will 
consist of a power of a multiplied into a power of x , hav- 
ing such a numeral coefficient prefixed as expresses the num- 
