will be equal to nothing: fince y=x-e. And confequently, by the laft article, the two laft terms of the transformed equation must vanish.

Suppose it is the cubic equation of § 33. that is propofed, viz. x3- px2 + qx-ro; and because we suppose x = e, therefore the last term of the transformed equation, viz. e3 — pe2 + qe - will vanifh. And fince two values of y vanish, the last term but one, viz. 3e'y 2 pey

+ qy, will vanish at the same time. So that 3e2 2 peqo. But, by fuppofition, ex, therefore, when two values of x, in the equation x3 · px2 + qx — r = o, are equal, it follows, that 3x2- 2px + q = 0. And thus, "the propofed cubic is depreffed to a quadratic that has one of its roots equal to one of the roots of that cubic."

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If it is the biquadratic that is proposed, viz. 304 px3 + qx2 rx + s = 0, and two of its roots be equal; then fuppofing ex, two of the values of y must vanish, and the equation of $34. will be reduced to this form,

4e3 - 3pe2 + 2 ger = 0; or because xe,

4x3- 3px2 + 2qxr = o..

=

In general, when two values of x are equal to each other, and to e, the two laft terms of the transformed equation vanish: and confequently, if you multiply the terms of the propofed equation by the indices of x in each term, the quantity that will arife will be = 0, and will give an equation of a lower dimension than the proposed, that shall have one of its roots equal to one of the roots of the proposed equation."

That the last two terms of the equation vanish when the values of x are fuppofed equal to each other, and to e, will alfo appear by confidering, that fince two values of y then become equal to nothing, the product of the values of y muft vanish, which is equal to the laft term of the equation; and because two of the four values of y are equal to nothing, it follows alfo that one of any three that can be taken out of these four must be = 0; and therefore, the products made by multiplying any three muft vanish; and confequently the coefficient of the last term but one, which is equal to the fum of these products, muft vanish.

§ 37. After the fame manner, if there are three equal roots in the biquadratic + - px3 + qx2 ➡ rx + s = o, and if e be equal to one of them; three values of y (xe) will vanish, and confequently y3 will enter all the terms of

M 3

the

the transformed equation; which will have this

form,

бег

6x2

--

34 + 4ey

-py3

}

***=0. So that here

3pe + q = 0; or, fince ex, therefore 3px + q = 0: and one of the roots of this quadratic will be equal to one of the roots of the propofed biquadratic.

In this cafe, two of the roots of the cubic equation 4x3- 3px2 + 2qx − r = 0 are roots of the proposed biquadratic, because the quantity 6x2 3px + q is deduced from 4x3-3px2 + 2qxr, by multiplying the terms by the indexes of x in each term.

In general," whatever is the number of equal roots in the propofed equation, they will. all remain but one in the equation that is de'duced from it by multiplying all the terms by the indexes of x in them; and they will all remain but two in the equation deduced in the fame manner from that," and fo of the reft.

§ 38. What we obferved of the coefficients of equations transformed by fuppofing y=xe, leads to this eafy demonstration of this Rule; and will be applied in the next chapter to demonstrate the Rules for finding the limits of equations.

It is obvious however, that though we make ufe of equations whofe figns change alternately, the fame reasoning extends to all other equations.

It is a confequence alfo of what has been demonftrated, that "if two roots of any equation, as x3- · px2 + qx − r = o, are equal, then multiplying the terms by any arithmetical feries, as a + 3b, a + 2b, a + b, a, the product will be = o."

For fince ax3 apx2 + aqx-aro; and 3x2 - 2px + q xbx = o, it follows that ax3 + 3bx3 — apx2 — 2bpx2 + aqx + bqx — ar = 0. Which is the product that arifes by multiplying the terms of the proposed equation by the terms of tlie feries, a + 3b, a + 2b, a + b, a; which may represent any arithmetical progreffion.

СНАР. V.

OF THE LIMITS OF EQUATIONS.

$ 39. W difcover the limits of the roots of

E now proceed to fhew how to

equations, by which their folution is much facilitated.

Let any equation, as x3- px + qx — r = 0, be proposed; and transform it, as above, into the equation,

Where the values of y are less than the respec tive values of x by the difference e. If If you fuppofe e to be taken fuch as to make all the coefficients, of the equation of y, positive, viz. e3 — pe2 + qe — r, 3e2 — 2pe + 9, 3e - p; then there being no variation of the figns in the equation, all the values of y must be negative; and confequently, the quantity e, by which the values of x are diminished, must be greater than the greatest positive value of x: and confequently must be the limit of the roots of the equation 23 - px2 + qx — r = 0. 23

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