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and from to t; and the remaining roots are negative. The rule also may be demonstrated.

Note. The impossible roots in this rule are suppofed to be either positive or negative.

In this example of a numeral equation x4—10x +35x2–50x+2450, the roots are, +1, +2, +3, +4, and the preceding observations with regard to the signs and coefficients take place.

Cor. If a term of an equation is wanting, the positive and negative parts of its coefficient must then be equal. If there is no abfolute term, then some of the roots must be =0, and the equation may be depressed by dividing all the terms by the lowest power of the unknown quantity in


of them. In this case also, x-O=0, x-0=0, be considered as so

of the component simple equations, by which the given equation being divided, it will be depressed so many degrees.


&c. may





Of the Transformation of Equations.


HERE are certain transformations of

equations necessary towards their solution; and the most useful are contained in the following propositions.

Prop. I. The affirmative roots of an equation become negative, and the negative become affirmative, by changing the signs of the alternate terms, beginning with the second.

Thus the roots of the equation **-*19x2+49x—30=o are +1, +2, +3,-5, whereas the roots of the equation, x4+x19x2—49x_-30=0,are-1, -2,-3, +5.


The reason of this is derived from the coinposition of the coefficients of thefe terms, which consist of combinations of odd numbers of the roots, as explained in the preceding chapter.

Prop. II. · An equation may be tranfformed into another that shall have its roots greater or less than the roots of the given equation by some given difference.

Let x be the unknown quantity of the equation, and e the given difference ; let y=x+e, then x=yFe; and if for x and its powers

in the given equation, yFe and its powers be inserted, a new equation will arise, in which the unknown quantity is y, and its value will be x +e; that is, its roots will differ from the roots of the given equation by e.

Let the equation proposed be * -px? + 9x_r=o, of which the roots must be diminished by e. By inserting for x and its

. powers, yte and its powers, the equation required is,

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Cor. I.


From this transformation, the second, or any other intermediate term, may be taken away ; granting the resolution of equations.

Since the coefficients of all the terms of the transformed equation, except the first, involve the powers of e and known quantities only, by putting the coefficient of term equal to o, and resolving that equation, a value of e may be determined; which being substituted, will make that term to vanish. Thus, in this example, to take


the fecond term, let its coefficient, 3emp=0, and e=ip, which being substituted for the new equation will want the second term. And universally, the coefficient of the first term of a cubic equation being i, and x being the unknown quantity, the second term may be taken away, by supposing x=yFip, +p being the coefficient of that term.


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Cor. 2. The second term may be taken away by the solution of a simple equation, the third by the solution of a quadratic, and so on. Cor.


If the second term of a quadratic equation be taken away, it will become a pure equation, and thus a solution of

quadratics will be obtained, which coincides with the solution already given in Part I.

Cor. 4. The last term of the transformed equation is the same with the given equation, only having e in place of x.

Prop. III. In like manner may an equation be transformed into another, of which the roots shall be equal :o the roots of the given equation, multiplied or divided by a given quantity:

Let » be the unknown letter in the given equation, and


that of the equation wanted; also let e be the given quantity.

To multiply the roots, let xe=y, and x=


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To divide the roots let * =y,andx=ye.


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