TRACT 28.

AND INFINITE SERIES.

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none at all; and consequently it has at least always one vealroot. Let that root he represented by r, which may be ei-ther positive or negative, and may be any one of the realroots, when none of them are imaginary: then, since anyone of the roots is equal to the sum of the other two, withtheir signs changed, the other two roots may be representedby ~ \r ± some other quantity, since the sum of thesetwo, with the signs changed, is = r. Now this supple-mental quantity, which is to be connected with — a r, bythe signs + and.—, to compose the other two roots, willbe a real quantity when those roots are real, but an imagi-nary one when they are imaginary, since the other part,

— a?', of those two roots, is real, by the hypothesis. Let

this supplemental quantity be represented by e, when it isreal, or e s /— 1 or ^/—e 2 , when it is imaginary: we shalluse the quantity e in what follows for the real roots; and itis evident, that by changing e for e s /— 1, or e z for — e%that is,- by barely changing the sign of e 2 wherever it isfound, the expressions will become adapted to the imaginaryroots. Hence then, the three roots are represented by r,and — J-r + e, and — — e; and consequently the three

equations, from whose continual multiplication by one an-other, the cubic equation is to be generated, will bex — r = 0, and x 4- \r — e = 0, and x + ir + e = 0.

9. Let now these three equations be multiplied together,and there will be produced this general cubic equation,wanting the second term, namely, x 3 x — r(J-r 2 — t 2 )

— 0, or x 3 x = r ( rr z — e 2 ), having three real roots; and

if the sign of e z be changed from — to +, it will then re-present all the cases which have only one real, and twoimaginary roots: and from the bare inspection of this equa-tion the following properties are easily drawn.

10. First, we hence find, that when the equation has three

real roots, the sign of the second term is always — ; forthe coefficient of that term, or p, is = — — <r, which is

always negative, when r and e are real quantities. And

VOL. II. G