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ON CUBIC EQUATIONS

TRACT 2?,

wanting the second term, where p and q represent anynumbers, positive or negative.

5. Now from the premises it follows, that this equationhas three roots; that some are positive, and others nega-tive; that two of them are of one affection, and are toge-ther equal to the third of a contrary affection, namely, ei-ther two negative roots, which are together equal to theother positive, or two positive roots equal to the third ne-gative.

6. But the signs of the three roots are easily known fromthe sign of the quantity q; the sign of the greatest rootbeing the same with the sign of q, when this quantity is onthe right-hand side of the equation, and the other two rootsof the contrary sign. For when q is on the same side ofthe equation with the other terms, it has been observed,that it is always equal to the continual product of all theroots, with their signs changed ; consequently q is equal tothe product of all the roots under their own signs, whenthat quantity is on the other or right-hand side of the equa-tion : but the product of the two less roots is always posi-tive, because they are of the same affection, either both -f-or both —; and therefore this product, drawn into the thirdor greatest root, will generate another product, equal to q,and of the same affection with this root.

7. But the roots of equations of the above form, are notonly positive, negative, or nothing, but sometimes alsoimaginary. We have found that the greatest root is posi-tive when q is positive, and negative when q is negative; asalso, that one root is = to 0 when q js = 0, and in this casethe other two roots must be equal to each other, with con-trary signs. But to discover the cases in which the equationhas imaginary roots, as well as many other properties of theequation, it will be proper to consider the generation of itas follows.

8. The roots of equations becoming imaginary in pairs,the number of imaginary roots is always even; and there-fore the cubic equation has cither two imaginary roots, or