The product of these two expressions will be a series in ascending powers of x; denote it by
;
then it is clear that A, B, C,\ldots are functions of m and n, and therefore the actual values of A, B, C,\ldots in any particular case will depend upon the values of m and n in that case. But the way in which the coefficients of the powers of x in (1) and (2) combine to give A, B, C,\ldots is quite independent of m and n; in other words, whatever values m and n may have A, B, C, \ldots preserve the same invariable form. If therefore we can determine the form of A, B, C,\ldots for any value of m and n, we conclude that A, B, C, \ldots will have the same form for all values of m and n.
The principle here explained is often referred to as an example of “the permanence of equivalent forms”; in the present case we have only to recognize the fact that in any algebraic product the form of the result will be the same whether the quantities involved are whole numbers, or fractions; positive, or negative.
We shall make use of this principle in the general proof of the Binomial Theorem for any index. The proof which we give is due to Euler.
533. To prove the Binomial Theorem when the index is a positive fraction.
Whatever be the value of m, positive or negative, integral or Fractional, let the symbol f(m) stand for the series
\ldots
then f(n) will stand for the series
\ldots
If we multiply these two series together the product will be another series in ascending powers of x, whose coefficients will be unaltered in form whatever m and n may be.
