if the last term has the sign +, since, whether we make x = 0, or x = M, we have always a positive result. But when this term is negative, we find, by making

་

three results, affected respectively with the signs +, -, and +, and, consequently, the proposed equation has, at least, two real roots in this case, the one positive, found between M and 0, the other negative, between 0 and -M; therefore, every equation of an even degree, the last term of which is negative, has at least two real roots, the one positive and the other negative.

215. I now proceed to the resolution of equations by approximation; and in order to render what is to be offered on this subject more clear, I shall begin with an example.

Let there be the equation

004 4 x 3

3x+27=0;

the greatest negative coefficient found in this equation being — 4, it follows (212), that the greatest positive root will be less than 5. Substituting for y x, we have

3

y1 + 4y3 + 3y + 27 = 0;

and as all the terms of this result are positive, it appears, that y must be negative; whence it follows, that x is necessarily positive, and that the proposed equation can have no negative roots; its real roots are, therefore, found between 0 and + 5.

The first method, which presents itself for reducing the limits, between which the roots are to be sought, is to suppose successively

x = 1, x = 2, x = 3, x = 4;

and if two of these numbers, substituted in the proposed equation, lead to results with contrary signs, they will form new limits to the roots. Now if we make

x = 1, the first member of the equation becomes + 21,

it is evident, therefore, that this equation has two real roots, the one found between 2 and 3, and the other between 3 and 4. To approximate the first still nearer, we take the number 2,5, which occupies the middle place between 2 and 3 (Arith. 129), the present limits of this root; making then x = 2,5, we arrive at the result

+ 39,0625

62,5 7,5 +27=- 3,9375;

as this result is negative, it is evident, that the root sought is between 2 and 2,5. The mean of these two numbers is 2,25; taking a = 2,3 we have the root sought within about one tenth of its value, and shall approximate the true root very fast by the following process, given by Newton.

We make x = 2,3+y; it is evident, that the unknown quantity y amounts only to a very small fraction, the square and higher powers of which may be neglected; we have then

(2,3) + 4 (2,3)3 y 4 (2,3)3 12 (2,3)3 y

3 (2,3)

substituting these values, the proposed equation becomes - 0,583917,812 y = 0,

Stopping at hundredths, we obtain for the result of the first operation

-

y=I 0,03 and x = 2,3 0,03 = 2,27. To obtain a new value of x, more exact than the preceding, we suppose x = 2,27+y; substituting this value in the proposed equation and neglecting all the powers of y' exceeding the first, we find

and, consequently, a 2,2675. We may, by pursuing this process, approximate, as nearly as we please, the true value of x. If we seek the second root, contained between 3 and 4, by the same method, we find, stopping at the fourth decimal place,

x = 3,6797.

216. We may ascertain the exactness of the method above explained, by seeking the limit to the values of the terms, which are neglected.

If the proposed equation were

xm + Pxm-1+Qxm

......

+ Tx+U= 0,

substituting a + y for x, we should have for the result the first of the equations found in art. 204, because a being not the root of

the equation, but only an approximate value of x, cannot reduce to nothing the quantity

am + Pam-1+ Q am-2

+ Ta + U.

Representing this last by V, we have, instead of the equation (d) above referred to, the following

Neglecting the powers of y exceeding the first, we have

and this value differs from the real value of y by

If a differs from the true value of x only by a quantity less

1

thana, the above mentioned error becomes less than that,

1

which would arise from putting-a in the place of y, which would

give

P

Finding the value of this quantity, we shall be able to determine, whether it may be neglected when considered with reference to V and if it be found too large, we must obtain for a a number,

which approaches nearer to the true value of x.

To conclude, when we have gone through the calculation with several numbers, y, y', y', &c. if the results thus obtained form a decreasing series, an approximation is certain.

217. The method we have employed above, is called the Method by successive Substitutions. Lagrange has considerably improved it.* He has remarked, that by substituting only entire

* See Résolution des Equations numériques.

numbers, we may pass over several roots without perceiving them. In fact, if we have, for example, the equation

by substituting for a the numbers, 0, 1, 2, 3, &c. we shall pass and, without discovering that they exist; for

over the roots

we shall have

It will be readily perceived,

results affected by the same sign. that this circumstance takes place in consequence of the fact, that the substitution of 1 for x changes at the same time the signs of both the factors, x, and, which pass from the negative state, in which they are when O is put in the place of x, to the positive; but if we substitute for r a number between and, the sign of the factor — alone will be changed, and we shall obtain a negative result.

We shall necessarily meet with such a number, if we substitute, in the place of x, numbers, which differ from each other by a quantity less than the difference between the roots and 1. If, for example, we substitute, 4, 4, 4, 4, &c. there will be two changes of the sign.

3

It may be objected to the above example, that when the fractional coefficients of an equation have been made to disappear, the equation can have for roots only either entire or irrational numbers, and not fractions; but it will be readily seen, that the irrational numbers, for which we have, in the example, substituted fractions for the purpose of simplifying the expressions, may differ from each other by a quantity less than unity.

In general, the results will have the same sign, whenever the substitutions produce a change in the sign of an even number of factors.* To obviate this inconvenience we must take the numbers to be substituted, such, that the difference between the smallest limit and the greatest, will be less than the least of the differences, which can exist between the roots of the proposed equation; by this means the numbers to be substituted will necessa

* Equal roots cannot be discovered by this process, when their number is even; to find these we must employ the method given in art. 205.

rily fall between the successive roots, and will cause a change in the sign of one factor only. This process does not presuppose the smallest difference between the roots to be known, but requires only, that the limit, below which it cannot fall, be determined.

In order to obtain this limit, we form the equation involving the squares of the differences of the roots (208).

Let there be the equation

to obtain the smallest limit to the roots, we make (212) z =

or, reducing all the terms to the same denominator,

r

1

and if represent the greatest negative coefficient found in this

น

It is only necessary to consider here the positive limit, as this alone relates to the real roots of the proposed equation.

less than the square of the smallest difference between the roots of the proposed equation, we may find its square root, or at least, take the rational number next below this root; this number, which I shall designate by k, will represent the difference which must exist between the several numbers to be substituted. We thus form the two series,

from which we are to take only the terms, comprehended between the limits to the smallest and the greatest positive roots,