and Infinite Series. 391 
this product, drawn into the third or greateft: root, will 
generate another product equal to q , and of the fame 
affection with this root. 
7. But the roots of equations of the above form are 
not only pofitive, negative, or nothing, but fometimes 
alfo imaginary. We have found that the greateft root is 
pofitive when q is pofitive, and negative when it is nega- 
tive; as alfo that one root is = to o when q is = o, and in 
this cafe the other two roots mult be equal to each other, 
with contrary figns. But to difeover the cafes in which, 
the equation has imaginary roots, as well as many other 
properties of the equation,, it will be proper to confider 
the generation of it as follows. 
8. The roots of equations becoming imaginary in. 
pairs, the number of imaginary roots is always even; 
and therefore the cubic equation has either two imagi- 
nary roots, or none at all; and confequently it has at 
leafl one real root. Let that root be reprefented by r, 
which may be either pofitive or negative, and may be 
any one of the real roots, when none of them are ima- 
ginary: then fince any one of the roots is equal to the 
fum of the other two with their figns changed, the other 
two roots may be reprefented by — \r ± fome other 
quantity, fince the fum of thefe two, with the figns 
changed, is = r. Now this fupplemental quantity, which 
is 
