TRACT XXVIIL

OF CUBIC EQUATIONS A&D INFINITE SERIES. READ At AMEETING OF THE ROYAL SOCIETY , JUNE 1, 1780.

The following pages are not to be understood as intendedto contain a complete treatise on cubic equations, with allthe methods of solution that have been delivered by other■writers; but they are chiefly employed on the improvementsof some properties that were before but partially known,with the discovery of several others, which appear to henew, and of no small importance! for I have only slightlymentioned such of the generally' known properties as werenecessary' to the introduction, or investigation, of the manycurious consequences herein deduced from them.

.Art. 1. Every equation, whose terms are expressed insimple integral powers, has as many? roots as there are di-mensions in the highest power. And when all the terms arebrought to one side of the equation, and the coefficient ofthe first term, or highest power, is 1, then the coefficientof the second term, is equal to the sum of all the roots withcontrary signs: the coefficient of the third term, is equal tothe sum of all the products made by multiplying every twoof the roots together; the coefficient of the' fourth term,equal to the sum of all the products arising from the multi-plication of every three of the roots together, &c; andthe last teim, equal to the Continual product of all theroots; the signs of all of them being supposed to bechanged, into the contrary signs, before these multiplica-tions are made. All this is evident from the generation ofequations. And, from these properties of the coefficients,the following deductions are easily made.