of the Binomial Theorem. 30 5, 
ther, and as it was proved in the second article that each of the 
quantities a, b, c , &c. must be found the same number of times 
in this product, if we can compute the number of times any 
one of those quantities enters into the coefficient of any term 
of the last equation, we shall then know how often each of 
the other enters into the same coefficient : and this may be 
done with ease, if of the quantities a, b, c, See. we fix upon that 
used in the last multiplication. For the last equation, and in- 
deed any other, may be considered as made up of two parts ; 
the first part being the equation immediately before the last 
multiplied by x, according to the 5th article, and the other 
being the same equation multiplied by the quantity adjoined 
to a: by the sign -js last used in the multiplication, according 
to the 6th article. This last used quantity, therefore, never en- 
ters into the members of the coefficient of the first of these two 
parts, but it enters into all the members of the coefficients of 
the last of them. But that part into which it does not enter 
has the same members as the coefficients of the equation im- 
mediately before the last, by the 5th article ; and when the 
members of the first part are multiplied by the last used quan- 
tity, the product becomes the second part of-the whole coeffi- 
cient above mentioned. 
Thus the first part of the cubic equation, by the 5th article 
is, X ^ ^ } x 2 -\- a b x, and as these coefficients are the same as 
the coefficients in the quadratic equation, being multiplied by 
c, and arranged according to the 6th article, we have the co- 
efficients of the second part of the cubic, viz. c 4 - a e . 
l 4 - a b c. 
-f b c ' 
Hence it is evident, that there are as many members in any 
