126 A Treatise js Algebra.

*3

ab acs-}- ad -j- be ?- jf 2 -f-bd -j- cd J

— abc-—abd

— aed

— bed

x -f- a bcd — S,

has four, s, b, c and d. Now from these Equations it isalso observable, that the Coefficient of the second Termis always equal to the Sum of all the Roots, with con-trary Signs; that the Coefficient of the third Term isalways equal to the Sum of their Rectangles, or of allthe Products that can possibly arise by combining them,two and two ; that the Coefficient of the fourth is equalto the Sum of all their Solids, or of all the Productswhich can possibly arise, by combining them three andthree, and that the last Term of all is produced by mul-tiplying them continually into each other. And allthis, it is manifest, will hold equally, when only someof the Roots are positive and the rest negative (due re-gard being had to the Signs.) Thus in the cubic Equa-

. -- -

tion a- — a xx — ix* + c = o, or *3 -|-

+ abl

—ac ? x 4- abc — o, where two of the Roots (a, b) are— be J

positive, and the other (— r) is negative, the Coefficientof the second Term is — a — b c, and that of thethird, ab — ac — be , or ab -|- a x — c bx— c, accord-ing to the preceding Observations. Hence it follows,that, if one of the Roots of an Equation be given, theSum of all the rest will likewise be given; and that inevery Equation where the second Term is wanting, theSum of all the negative Roots is exactly equal to that ofall the positive ones ; because, in this Cafe, they mu-tually destroy each other. But when the Coefficientof the second Term is positive, then the negativeRoots, taken together, exceed the positive ones. Butthe negative Roots, in any Equation, will -be changedto positive ones, and the positive to negative, by chang-ing the Signs of the second, fourth and sixth Terms,and so on, alternately. Thus, the foregoing Equation