Mr. Robertson’s new Demonstration 
31B 
plain axiom, viz. that equals being multiplied by equals the 
products are equal. 
In the second place it is to be observed, that the whole 
coefficient of any power of x, in the products of multiplication 
A, may be reduced to the regular binomial form, established 
in the 13th article. Thus n . ~~ -\-mn-\-m . 1 , the whole 
coefficient of x\ by actual multiplication becomes 
2 
— n-\-m 
n-\-m — 1 
2 
. n-\-m 
m — 1 
A1 n 1 «— 2 I 
Also 11 . . h mil . 
2 3 1 
, the whole coefficient of 
x\ by actual multiplication becomes +m — + 3 ” m -{- 
. 3 ” = _ 2 ^= 2 . And from the 
6 '23 
preceding observation it is evident, that we may in the same 
manner, reduce the whole coefficient of any other power of 
x, in the products of multiplication A to the regular binomial 
form. 
16. But in proceeding, as above, to change the form of the 
coefficients prefixed to any power of x, in multiplication A, 
into the regular binomial form, we are not under the necessity 
of supposing n and in to be whole numbers. The actual mul- 
tiplications will end in the same powers of 11 and m, the same 
combinations of them, and the same numerals, whether we 
consider n and m as whole numbers or as fractions. 
We are therefore at liberty to suppose n and m to be any 
two fractions whatever, in the two series multiplied into one 
another in multiplication A, and the same two fractions will 
take the place of n and m respectively in the regular binomial 
series i-^n^mx-^n-^m 
n-i-m — 1 
x^n^m . 
n+m—z^ 
1 M 1 ' """'"X 
2 
2 
3 
