which form is general, whatever may be the different signs of the roots, or of the terms of the equation; taking a, b, c, &c. as well as A, B, C, &c. in+or-as they may happen to be.

534. Hence, since the two equations (1), (3), are identical, the coefficients of the like powers of x, (Art. 435), are equal; and, consequently, the following relations between the coefficients and roots will be sufficiently obvious.

1. The sum of all the roots of any equation, having its terms arranged according to the order of the powers of the unknown quantity, is equal to the coefficient of the second term of that equation, with its sign changed.

II. The sum of the products of all the roots, taken two and two, is equal to the coefficient of the third term, with its proper sign; and so on.

III. The continued product of all the roots, is equal to the last term, taken with the same or a contrary sign, according as the degree of the equation is even or odd.

535. It is very proper to observe, that we cannot have all at once x=a, x=b, a=c, &c. for the roots of any equation, as in the formula (2); except when a=b=c=d, &c., that is, when all the roots are equal. The factors x-a, x-b, x-c, &c. exist in the same equation; because algebra gives, by one and the same formula, not only the solution of the particular problem from which that formula may have originated; but also the solution of all problems which have similar conditions. The different roots of the equation satisfy the respective conditions; and those roots may differ from one another by their quantity, and by their mode of existence,

536. To this we may likewise add, that, if the roots of any equation be all positive, as in formula (2), where the factors are of the form

...

(x-a) (x-b) (x-c) (x−d) ・・・ (x−1)=0, the signs of the terms will be alternately + and -; as will readily appear from performing the operation required.

537. But if the roots be all negative, in which case the factors will be of the form

(x+α) (x+b)(x+c) (x+d) (x+1)=0, the signs of all the terms will be positive; because the equation arises wholly from the multiplication of positive quantities.

Some equations have their roots in part positive, in part negative: Thus, in the cubic equation, (x-a)x(x—b)×(x+c)=0, or x2+(c-a-b) x2+ (ab-ac-bc) xx+abc=0, there are two positive and one negative root; because, when x-a=0, x=a; x-b=0, x=b; x+c=0, x=-c.

538. Any equation, having fractional coefficients, may be transformed into another, that shall have the coefficient of its first term unity, and those of the rest, as well as the absolute term, whole numbers.

For let there be taken, instead of a general equation of this kind, the following partial example,

x 3 + — x2 + 3 x+3=0,

which will be sufficient to show the method that should be followed in other cases.

3

Then if each of the terms be multiplied by the product of the denominators, or by their least common multiple, we shall have 12x+6x2+8x+9=0, where the coefficients and absolute term are all whole numbers.

y

And if 12x, in this case, be put=y, or x=; x=12, there will arise by substitution,

Which last equation, when all its terms are multiplied by 122, gives y3+6y+96y+1296=0; where

the coefficient of the first term is unity, and those of the rest whole numbers, as was required.

So that when the value of y in this equation is known, we shall have for the proposed equation

539. Any equation may be transformed into another, the roots of which shall be greater or less than those of the former by a given quantity.

Thus, let there be taken, as before, the following general equation,

m

x+Ax+Bx-2+Сx-3+.. Tx+V=0. And suppose it were required to transform it into another, whose roots shall be greater than those of the given equation by e.

Then, if y be made to represent one of these roots, we shall have, by the nature of the question, y=x+e, or x=y-e.

And, consequently, by substituting y-e for x, in the proposed equation, there will arise

which equation will evidently fulfil the conditions required, y being here greater than a by e. And if y be taken =x-e, or x=y+e, we shall obtain, by a similar substitution, an equation whose roots are less than those of the given equation by e.

540. Whence, also, as e, in the above case, is indeterminate, this mode of substitution may be used for destroying one of the terms of the proposed equation. For putting in the above expression the coefficient-me+A=o, we shall have

where it is plain, that the second term of any equation may be taken away, by substituting for the unknown

quantity some other unknown quantity, together with such a part of the coefficient of the second term, taken. with a contrary sign, as is denoted by the index of the highest power of the equation.

Thus, for example, to transform the equation x3-9x2+7x+12=0 into one which shall want the second term. Assume x=y+3; then

x3=y3 +9y2+27y+27)

-9x2-9y2-54y-81

+7x=

+12 =

3

+7y+21

=0,

+12

that is, y3-204-21=0; and if the values of y be a, b, c, the values of x are a+3, b+3, and c+3. The third term of the proposed equation may also be taken away by means of the coefficient, or formula,

where the determination of e requires the solution o an equation of the second degree; and so on.

541. Any proposed equation may be transformed into another, the roots of which shall be any multiples or parts of those of the former.

Thus, let there be taken, as in the former propositions, the general equation

x+Axm-1+Bx-2+Сx-3+.. Tx+V=0.(1). And, in order to convert it into another, whose roots shall be some multiple of those of the given

equation, let there be put y=ex, or x=.

e

Then, by substituting this value for x in the proposed equation, there will arise

ym--2 em--2

y ym em em--1

+A +B +

And, consequently, if this be multiplied by em, we

shall have

y+Aey+Be2ym-+.... Tem-1y+Ve=0, which equation will evidently fulfil the conditions required, y being equal to ex.

x

And if y be put= or xey, we shall obtain, by a similar substitution of this value for x, and then dividing by em, the equation

T V

+
em--1 em

where the roots are equal to those of the proposed equation, divided by e.

And it may easily be proved, that if the alternate terms, beginning with the second, be changed, the signs of all the roots are changed.

542. For a more particular account of the general Theory and Doctrine of Equations, see BONNYCASTLE'S Algebra, vol. ii. 8vo. BRIDGE's Equations, and LAGRANGE's Traité de la Resolution des Equations Numeriques; where the intelligent reader will find a full investigation of this part of analysis.

§ II. RESOLUTION OF CUBIC EQUATIONS BY The rule OF CARDAN, OR OF SCIPIO FERREO.

543. Cubic equations, as has already been observed in Chap. VIII., are of two kinds; that is, pure and adfected. All pure equations of the third degree are comprehended in the formula a3n, where n may be any number whatever, positive or negative, integral or fractional. And the value of x is obtained, (Art. 396), by extracting the cube root of the number n.

544. But in this manner, we obtain only one value for x; whereas (Art. 531), every equation of the third degree has three values. In order to show how the two remaining values of x may be determined in equations of the above form, let us, for example, consider the equation x3-8=0; where x is readily found =2. And as 2 is a root of the proposed equa