TfeACt 28.

ON COBIC ECtUATIONS, &C.

79

2. If the signs of all the roots of an equation be changed,and another equation be generated from the same roots,with the signs so changed ; the terms of this last equationwill have the same coefficients as the former; only, the signsof all the even terms will be changed, but not those of theodd terms: for the coefficients of the second, fourth; andthe other even terms, are made up of products consistingeach of an odd number of factors ; while those of the third,fifth, and other odd terms are composed of productshavingan even number of factors: and the change of the signs ofall the factors produces a change in the sign of the continualproduct of an odd number of factors; but no change in thesign of that of an even number of factors. Therefore,changing the signs of all the even terms, namely, the se-cond, fourth, &c, produces no alteration in the roots, butonly in their signs, the positiv'e i*oots being changed intonegative, and the negative into positive. But by changingany or all the signs of the odd terms, the equation will nolonger have the same roots as before, but will have newroots of very different magnitudes from those of the former,unless the sign of the first term, or highest power, bechanged also; but this term is always to be supposed toremain positive.

S. It also follows, that when any term is wanting in an'equation, or the coefficient of any term is equal to 0, thesum of the negative products, in the coefficient of thatterm, is equal to the sum of the positive products in thesame. And if it he the second term which is wanting, thenthe equation has both negative and positive roots, and thesum of the negative roots is equal to the sum of the positiveones. But if it be the last term which is wanting, then oneof the roots of tire equation is equal to nothing. Andhence arises a method of transforming any equation intoanother which shall want the second term : and to this latterstate it will be proper to transform every cubic equation,before we attempt the solution of it.

4i Let therefore x 3 + p x = q be such a cubic equation,