If it were required to find the final equation in, we observe that x and y enter into the primitive equations under the same forms; hence, x may be changed into y and y into x, without destroying the equality of the members. Therefore,

4x6

is the final equation in x.

·6x4 + 3x2 — 1=0

2. Find the final equation in y, from the equations

x3 3yx2 + (3y2 — y + 1) x − y3 + y2 —2y = 0,

x2 - Qyx + y2 - y = 0.

First Operation.

x3 — 3yx2 + (3y2 − y + 1) x − y3 + y2 — 2y||x2 — Qxy + y2 — y

Second Operation.

x-y

is the final equation in y. This equation gives

y = 1 and y = 0.

Placing the preceding remainder equal to zero, and substi tuting therein the values of y,

from which the given equations may be entirely solved.

CHAPTER XI.

CONTAINING BUT ONE

SOLUTION OF NUMERICAL EQUATIONS UNKNOWN QUANTITY.—STURM'S THEOREM.-CARDAN'S RULE.-HORNER'S METHOD.

275. THE principles established in the preceding chapter, are applicable to all equations, whether the co-efficients are numerical or algebraic. These principles are the elements which are employed in the solution of all equations of higher degrees.

Algebraists have hitherto been unable to solve equations of a higher degree than the fourth. The formulas which have beer deduced for the solution of algebraic equations of the higher degrees, are so complicated and inconvenient, even when they can be applied, that we may regard the general solution of an algebraic equation, of any degree whatever, as a problem more curious than useful.

Methods have, however, been found for determining, to any degree of exactness, the values of the roots of all numerical equations; that is, of those equations which, besides the unknown quantity, involve only numbers.

It is proposed to develop these methods in this chapter.

276. To render the reasoning general, we will take the equation,

X = xm + Pxm--1+ Qxm−2+ U = 0.

Q

in which P, denote particular numbers which are real, and either positive or negative.

If we substitute for x a number a, and denote by A what I becomes under this supposition; and again substitute a + u for x, and denote the new polynomial by A': then, u may be taken so small, that the difference between A' and A shall be less than any assignable quantity.

(Art. 264), become, when we make xa, we

It is now required to show that this difference may be rendered less than any assignable quantity, by attributing a value sufficiently small to u.

If it be required to make the difference between A' and A less than the number N, we must assign a value to u which will satisfy the inequality

Let us take the most unfavorable case that can occur, viz., let us suppose that every co-efficient is positive, and that each is equal to the largest, which we will designate by K. Then any value of u which will satisfy the inequality

K(u + u2 + u3 + . . . . um) < N

will evidently satisfy inequality (3).

(4),

Now, the expression within the parenthesis is a geometrical progression, whose first term is u, whose last term is um, and whose ratio is u; hence (Art. 188),

Substituting this value in inequality (4), we have,

N m

ber of inequality (5) reduces to N, and since

is less

than 1, the second factor is less than 1; hence, the first mem

ber is less than N.

We conclude, therefore, that u =

N
N+K'

and every smaller

value of u, will satisfy the inequalities (3) and (4), and consequently, make the difference between A' and A less than any assignable number N.

If in the value of A', equation (1), we make u=

is plain that the sum of the terms

Bu + Cu2+ Du3 + um

...

A

it

A+K'

will be less than A, from what has just been proved; whence we conclude that

In a series of terms arranged according to the ascending powers of an arbitrary quantity, a value may be assigned to that qun so small, as to make the first term numerically greater than the sum of all the other terms.

First Principle.

277. If two numbers p and q, substituted in succession in the place of x in the first member of a numerical equation, give results affected with contrary signs, the proposed equation has a real root, comprehended between these two numbers.

Let us suppose that p, when substituted for x in the first member of the equation

X=0, gives +R,

and that q, substituted in the first member of the equation X = 0, gives - R'.

Let us now suppose to vary between the values of p and by so small a quantity, that the difference between any two corresponding consecutive values of X shall be less than any assignable quantity (Art. 276), in which case, we say that X is subject to the law of continuity, or that it passes through all the intermediate values between R and

R'.

Now, a quantity which is constantly finite, and subject to the 'aw of continuity, cannot change its sign from positive to nega

1

tive, or from negative to positive, without passing through zero: hence, there is at least one number between p and q which will satisfy the equation

X = 0,

and consequently, one root of the equation lies between these numbers.

278. We have shown in the last article, that if two numbers be substituted, in succession, for the unknown quantity in any equation, and give results affected with contrary signs, that there will be at least one real root comprehended between them. We are not, however, to conclude that there may not be more than one; nor are we to infer the converse of the proposition, viz., that the substitution, in succession, of two numbers which include roots of the equation, will necessarily give results affected with contrary signs.

Second Principle.

279. When an uneven number of the real roots of an equation is comprehended between two numbers, the results obtained by substituting these numbers in succession for x in the first member, will have contrary signs; but if they comprehend an even number of roots, the results obtained by their substitution will have the same sign.

To make this proposition as clear as possible, denote by a, b, c, . . . those roots of the proposed equation,

X = 0,

which are supposed to be comprehended between p and q, and by Y, the product of the factors of the first degree, with reference to x, corresponding to the remaining roots of the given equation.

The first member, X, can then be put under the form,

(x − a) (x —b) (x — c) . . . x Y = 0.

Now, substituting p and q in place of x, in the first mem

ber, we shall obtain the two results,