312 Mr. Robertson's new Demonstration 
coefficients as the cubic : the only effect of this multiplication 
being the increase of the exponents of x by 1 . 
8. But when the same equation is multiplied by the quan- 
tity adjoined to x by the sign -}-> each term of the product, in 
order to rank under the same power of x, must be drawn 
one term back. Thus when the hrst term of the cubic is mul- 
tiplied by d, the product must be placed in the second term of 
the biquadratic. When the second term of the cubic is multi- 
plied by d, the product must be placed in the third term of 
the biquadratic: and so of others. 
9. As the equation last produced is the product of all the 
compound quantities x-\- a, x-\-b, x-{- c, &c. into one another, 
and as it was proved in the fourth 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, &c. we fix upon 
that used in the last multiplication. For the last equation, 
and indeed 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 7th article, and the 
second part being the same equation multiplied by the quan- 
tity adjoined to x by the sign -{> last used in the multiplication, 
according to the 8th article. This last used quantity, therefore, 
never enters 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 
