454. COR. 1. COR. 1. If any term in the given equation be wanting, the corresponding term will be wanting in the transformed equation; thus, if the original equation want the second term, the transformed equation will want the last term but one, &c. because the coefficients in the transformed equation are the coefficients of the original equation in an inverted order.

455. COR. 2. If the coefficients of the terms, taken from the beginning of an equation, be the same with the coefficients of the corresponding terms, taken from the end, with the same signs, the transformed will coincide with the original equation, and their roots will therefore be the same.

Let a, b, c, be roots of the equation

-1

x2 - pœ”-1 + q x2-2.....+qx2 - px + 1 = 0;

the transformed equation will be

2-1

y" — py”-1 + qyn-2. + qy2 — py + 1 = 0,

.....

and a, b, c, must also be roots of this equation; but the roots of this equation are the reciprocals of the roots of the original

equation; that is, the roots of either equation are

456. COR. 3. If the equation be of an odd number of dimensions, or if the middle term of an equation of an even number of dimensions be wanting, the same thing will hold when the signs of the corresponding terms, taken from the beginning and end, are different.

Ex. The roots of the equation 3 – px2 + px − 1 = 0, are

terms of the transformed equation be changed, it will coincide with the original equation; and by changing the signs of all the terms, we do not alter the roots. (Art. 444).

457. DEF. The equations described in the last two corollaries are called recurring equations.

458. COR. 4. One root of a recurring equation of an odd number of dimensions will be + 1, or 1, according as the sign of the last term is or +; and the rest will be of the

For if 1, in the former case, and

1, in the latter, be substituted for the unknown quantity, the whole vanishes; thus, if

and for a we substitute + 1, it becomes

1-p+q-q+p-1=0;

and it appears from Art. 453, that if a, b, c, &c. be roots of

459. To transform an equation into one whose roots are the squares of the roots of the given equation.

-2

1

3

x2 + qx2¬2 + sx2-1 + &c. = px2-−1 + rx2−3 + &c. ;

and by squaring both sides,

-2

2n-4

x2n + 2qx2n−2 + (q2 + 28).x2n−1 + &c. = p2x2n−2 + 2prx2n−a + &c.

and again by transposition,

x21 — (p2 - 2q). x2n−2 + (q2 − 2 pr + 28). x2n− − &c. assume y = x2, then

n-1

y" - (p2 - 2q). y^-1 + (g2 — 2pr + 28). y"-2 - &c. = 0,

in which equation the values of y are the squares of the values

of x.

460. COR. If the roots of the original equation be a, b, c, &c. then

p2 — 2q = a2 + b2 + c2 + &c.

2

q2 - 2pr + 28 a2b2 + a2c2 + &c.
− 2s =

(Art. 436).

Other transformations may be seen in Waring's Meditationes Algebraica, Prob. 5; and, indeed, whoever would fully understand the nature of equations must have recourse to that Work.

[The more modern writers on this subject are Stevenson and Hymers.]

LIMITS OF THE ROOTS OF EQUATIONS.

461. If a, b, c, d, &c. be the roots of an equation, taken in order of magnitude, that is, a greater than b, b greater than c, &c.* the equation is

(x − a) (x − b) (x − c) (x + d) . &c.

= 0;

in which, if a quantity greater than a be substituted for a, as every factor is, on this supposition, positive, the result will be positive; if a quantity less than a, but greater than b, be substituted, the result will be negative, because the first factor will be negative and the rest positive; if a quantity between b and c be substituted, the result will again be positive, because the two first factors are negative and the rest positive: and so Thus quantities which are limits to the roots of an equation, or between which the roots lie, if substituted successively for the unknown quantity, give results alternately positive and negative.

on.

462. Conversely, if two magnitudes, when substituted for the unknown quantity, give results affected with different signs, an odd number of roots must lie between them; and if a series of magnitudes, taken in order, can be found, which give as many results, alternately positive and negative, as the

* In this series, the greater d is, the less is-d. And whenever a, b, c, -d, &c. are said to be the roots of an equation taken in order, a is supposed to be the greatest. Also, in speaking of the limits of the roots of an equation, we understand the limits of the possible roots.

?

equation has dimensions, these must be limits to the roots of the equation; because an odd number of roots lies between each two succeeding terms of the series, and there are as many terms as the equation has dimensions; therefore this odd number cannot exceed 1.

463. If the results arising from the substitution of two magnitudes for the unknown quantity be both positive or both negative, either no root of the equation, or an even number of roots, lies between them.

464. COR. If m, and every quantity greater than m, when substituted for the unknown quantity, give positive results, m is greater than the greatest root of the equation.

465. To find a limit greater than the greatest root of an equation.

e, b

е, с

Let the roots of the equation be a, b, c, &c.; transform it into one whose roots are a e, &c. and if, by - trial, such a value of e be found, that every term of the transformed equation is positive, all its roots are negative (Art. 430), and consequently e is greater than the greatest root of the proposed equation.

Ex. 1. To find a number greater than the greatest root of the equation a3 - 5 x2+7x − 1 = 0.

in which equation, if 3 be substituted for e, each of the quantities,

e3-5e2+7e1, 3e2 - 10e +7, 3e-5,

is positive, or all the values of y are negative; therefore 3 is greater than the greatest value of x.

Ex. 2. In any cubic equation of this form,

every term of which is positive; therefore √q is greater than the greatest value of x.

466.

COR. If the signs of the roots be changed, a limit greater than the greatest root of the resulting equation, with its sign changed, is less than the least root of the proposed equation.

Ex. Required a limit less than the least root of the equation y3 3y+ 72 = 0.

When the signs of the roots are changed, this equation becomes y3-3y720. (Art. 445).

and, if 5 be substituted for e, every term becomes positive, consequently 5 is greater than the greatest root of the equation y3 — 3y — 72 = 0; and 5 less than the least root of the equa

tion y3-3y+72 = 0.

467. The greatest negative coefficient increased by unity is greater than the greatest root of an equation.

and if the coefficients be equal to each other,

that is, "-p ×

= 0,

+ x + 1) = 0,

x" - 1

= 0

(Art. 277).