entire numbers, we may pass over several roots without perceiving them. In fact, if we have, for example, the equation (x) (x-1) (x-3) (x-4)=0, by substituting for x the numbers 0, 1, 2, 3, &c. we shall pass over the roots and, without discovering, that they exist; for we shall have (0) (0) (0—3) (0-4) — (1 − 3) (1 − 1) (1 − 3) (1 - 4) = + × 1 × 2 × 3, results affected by the same sign. It will be readily perceived, 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 x , which from the negative state, in which they are when 0 is put in the place of a, to the positive; but if we substitute for x a number between and, the sign of the factor x-alone will be changed, and we shall obtain a negative result. pass 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. If, for example, we substitute 4, 4, 4, 4, 4, &c. there will be two changes of the sign. 2 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 necessarily fall * 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. 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 z"+pz"¬1+qz"-2....+tz + u = 0.... (D), to obtain the smallest limit to the roots, we make (212) ≈ = or, reducing all the terms to the same denominator, 1 1 + t = vn-2 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, and those to the smallest and the greatest negative roots of the proposed equation. Substituting these different numbers, we shall arrive at a series of results, which will show by the changes of the sign that take place, the several real roots, whether positive or negative. 218. Let there be, for example, the equation x3-7x+7=0, from which in art. 208 was derived the equation making = 1, and, after substituting this value, arranging the result with reference to v, we have v3 — 9 v2 + 43&v — 2=0, = fulfilled, if we make k; but it is only necessary to suppose k; for by putting 9 in the place of in the preceding equation, we obtain a positive result, which must become greater, when a greater value is assigned to v, since the terms and 9 v2 already destroy each other, and 43 v exceeds 11. The highest limit to the positive roots of the proposed equation7x+7=0, is 8, and that to the negative roots-8; we must therefore substitute for x the numbers 0, 1, 2, $ ....... 3, the differences between the several values of x' will be triple of those between the values of x, and consequently will exceed unity; we shall then have only to substitute successively The signs of the results will be changed between +4 and +5, between 5 and 6, and between 9 and 10, so that we shall have for the positive values, and the negative value of x' will be found between 9 and 10, that of a between and. Knowing now the several roots of the proposed equation within, we may approach nearer to the true value by the method explained in art. 215. 219. The methods employed in the example given in art. 215, and in the preceding article, may be applied to an equation of any degree whatever, and will lead to values approaching the several real roots of this equation. It must be admitted however, that the operation becomes very tedious, when the degree of the proposed equation is very elevated; but in most cases it will be unnecessary to resort to the equation (D), or rather its place may be supplied by methods, with which the study of the higher branches of analysis will make us acquainted.* I shall observe however, that by substituting successively the numbers 0, 1, 2, 3, &c. in the place of x, we shall often be lead to suspect the existence of roots, that differ from each other by á quantity less than unity. In the example, upon which we have been employed, the results are +7, +1, +1, +13, which begin to increase after having decreased from +7 to +1. From this order being reversed it may be supposed, that between the numbers +1 and 2 there are two roots either equal or nearly equal. To verify this supposition, the unknown quantity should be multiplied. Making x = we find y 10 ys, ·700 y +7000 = 0, an equation, which has two positive roots, one between 13 and 14, and the other between 16 and 17. The number of trials necessary for discovering these roots is not great; for it is only between 10 and 20, that we are to search for y; and the values of this unknown quantity being * A very elegant method, given by Lagrange for avoiding the use of the equation (D), may be found in the Traité de la Résolution des Equations numériques. determined in whole numbers, we may find those of x within one tenth of unity. 220. When the coefficients in the equation proposed for resolution are very large, it will be found convenient to transform this equation into another, in which the coefficients shall be reduced to smaller numbers. If we have, for example, we may make x=10%; the equation then becomes • ·8319,98 z2 - 14,937 +0,5 = 0. If we take the entire numbers, which approach nearest to the coefficients in this result, we shall have ≈ 823 +20%2-15% +0,5=0. It may be readily dicovered, that has two real values, one between 0 and 1, the other between 1 and 2, whence it follows, that those of the proposed equation are between 0 and 10, and 10 and 20. I shall not here enter into the investigation of imaginary roots, as it depends on principles we cannot at present stop to illustrate; I shall pursue the subject in the Supplement. 221. Lagrange has given to the successive substitutions a form, which has this advantage, that it shows immediately, what approaches we make to the true root by each of the several operations, and which does not presuppose the value to be known within one tenth. Let a represent the entire number immediately below the root sought; to obtain this root, it will be only necessary to augment a by a fraction; we have therefore x = a + The equation 1 y involving y, with which we are furnished by substituting this value in the proposed equation, will necessarily have one root greater than unity; taking b to represent the entire number immediately below this root, we have for the second approximation x=a+. But b having the same relation to y, which a has 1 1 to x, we may, in the equation involving y, make y = b+7, and y' will necessarily be greater than unity; representing by b' the entire number immediately below the root of the equation in y', we have y= b + 1 = bb + 1 b b' |