464 
ON THE BINOMIAL THEOREM, 
[533 
It is to be observed that there is not in the demonstration any employment of 
the so called “ principle of equivalent forms,” we do not from the truth of an equation 
for positive integer values of to, n, infer the truth of it for any values whatever of 
to, n ; there is the intermediate step, that being true for integers, it is true identically ; 
and this identical truth of the equation depends on the circumstance that comparing 
on the two sides of the equation the coefficients of the successive powers x°, x\ x 2 , &c., 
these coefficients are in every case finite, rational, and integral functions of to, n. 
For instance, comparing the coefficients of x 2 , we have 
(to + n) (to + n — 1) = to (to — 1) + 2mn + n (n — 1), 
and any such equation, being true for all positive integer values of to, n, will be 
true identically ; developing the two sides, the equation is in fact 
to 2 + 2 mn + n 2 — m —n =m 2 —m + 2 mn + 11 2 — n. 
The reasoning is thus perfectly good ; but I remark that it is quite as easy to prove 
the general equation of which the last mentioned equation is an example, as it is to 
prove the binomial theorem for positive integer indices ; and consequently that we can 
withmb the aid of the binomial theorem for positive integer indices prove the fundamental 
equation 
jl + mx + — x ~ + &c -| j 1 + nx + + &c ’| 
_ m + n (m + n) (m + n — 1) „ , p 
= n—— x + 1 i 2 ' x + &c - 
To show this I introduce the factorial notation and write 
m (m — 1) ... (m — r+ 1) = [mf ; 
this being so, the equation obtained by comparing the coefficients of x r is readily found 
to be 
[m + nf = [m] r + j [mf- 1 [yi] 1 + r [mf- 2 [w] 2 + &c., 
and I say that this, the factorial binomial theorem for a positive integer index r, is 
proved as easily and in the same manner as the binomial theorem for a like value 
of the index ; or say as the equation 
(m + n) r — nf + j m' ,_1 n 1 + r m r ~ 2 n 2 + &c. 
To show how this is, I form the values of [m + n] 1 , [m + nf [m + nf, &c. successively, 
by what may be termed the process of varied multiplication : we have 
[to + n] 1 = m + n;