533] 
463 
533. 
ON THE BINOMIAL THEOREM, FACTORIALS, AND DERIVATIONS. 
[From the Oxford, Cambridge, and Dublin Messenger of Mathematics, vol. v. (1869), 
pp. 102—114.] 
The following was part of my course of lectures in the year 1867. 
The proof commonly called “ Euler’s ” of the binomial theorem is as follows: the 
theorem is assumed to be true for positive integer indices; that is, it is assumed that 
for any positive integer m we have 
(1 + x) m = 1 + mx + — 
This being so, since (1 + x) m . (1 + x) n = (1 + x) m+n , the equation 
m (m — 1) „ 
|1 + mx H y~2 
x- + &c. H1 + nx + 
n (n — 1) 
1.2 
X 1 + &c. 
, m + n (m 4- n) (m + n — 1) , „ 
= 1 + —x + ^ L x 1 + &c. 
is true for any positive integer values whatever of the indices m, n; the equation is 
therefore true identically; and it is consequently true for all values whatever of the indices 
m, n. But any function <^m of m, satisfying the functional equation frm. <j>n (m + n), 
is an m th power, = C m suppose; that is, we have 
C m = 1 + mx + 
m (m — 1) 
1.2 
x 1 + &c., 
where C is a constant, viz. it is independent of m; but the value of C will of course 
depend upon x; and if, in order to determine it, we write m= 1, the equation gives 
C — 1 + x; that is, we have 
(1 + x) m = 1 + mx + m 9 ~— x 1 + &c., 
which is the binomial theorem in its general form.