119] 
ON A THEOREM FOR THE DEVELOPMENT OF A FACTORIAL. 
101 
A preceding formula gives at once the theorem 
/4(0, = 
It may be as well to remark, with reference to a demonstration frequently given 
of the binomial theorem, that in whatever way the binomial theorem is demonstrated 
for integer positive indices, it follows from what has preceded that it is quite as easy 
to demonstrate the corresponding theorem for the factorial \m\ p . But the theorem 
being true for the factorial _ \ni\ p , it is at once seen that the product of the series 
for (1 + x) m and (1 + x) n is identical with the series for (1 + x) m+n , and thus it becomes 
unnecessary to employ for the purpose of proving this identity the so-called principle 
of the permanence of equivalent forms; a principle which however, in the case in 
question, may legitimately be employed.