If, therefore, we can find a value of a such that N is equal to, or greater than, Κα 1 we shall have effected the proof of the proposition. Now if x = we shall have Κα N N+K' = N; and as this is always greater than all the terms following N in the given equation, the condition is entirely fulfilled. -X THEOREM XII. The consecutive roots of ƒ1 (x) = 0 lie each in succession between the consecutive roots of f(x) = 0. Let r1, r2,....r, be the roots of ƒ (x) = Ax2 + Bä”−1 + = 0, ... and 1, P2. P-1, be those of ƒ1 (x) = nAx”—1 + (n−1) Bæ”−2+ = 0. Reduce the roots of ƒ (x) = =0 by the indeterminate quantity r, (prob. II.) which will give the roots of the transformed equation respectively equal to r1 r2r,.... r—r. But this transformation gives M1 = ƒ,(r), and N1 =ƒ(r). Now the value of M, in this reduced equation is (theor. VI.) M1 = (r—r,) (r—r2) (r−r3)... • (r—r2~1) + (r−r,) (r—rg) (r—r1)........(r—r2) + (r−r2) (r−r3) (r—r1). • • . (r—r2) 1 r, Now in this expression there is but one group of factors from which any one of the given factors is absent, as, for instance, r-r1. If then in M1 or ƒ (r) we give the indeterminate quantity r the successive values r1, ̃„.... ̃„, the several results will comprehend only one set of factors, viz. that in which r1, ̃1⁄2‚..........r are thus rendered successively absent; and we will suppose them so ranged that TM1, ̃1⁄2, . . . .”, are in the order of descending values. This will give (r2—r1) (r2−r3) • • • • (T2—r2) = · (rı—r2) (rı−r3) • • • • (~1—rn) = +K, since all the factors are +. =—к, since one factor only is (T3—r1) (r3—r2). • • • (T3—rn) = + «, since two factors only are —. And so on through the entire series of results. But when a series of quantities r1, r1⁄2, .... r, are substituted in an equation ƒ1 (x) =0, which give results alternately + and —, there is in all cases one root of the equation ƒ1 (x) = 0 comprehended between those numbers (theor. IX.) But P1, P2, P-1 are the roots of f(x) = 0; and hence these values lie between T1, T2, .r the roots of ƒ (x) = 0. That is, the roots of both equations being ranged in the order of descending magnitude follow each other thus: .... .... Cor. In the same way it may be shown, that the roots of f2 (a) = 0 lie between those of f(x): =0; those of ƒ ̧ (x) = 0 between those of ƒ2 (x) = 0, and so on. Scholium. These properties admit of various applications in the higher theory of algebraic equations, and are popularly known as the limiting equations of Newton. In actual numerical solution, however, they are now of little use; and they are only given here in justification of one or two processes employed. The equations are evidently the same which have been before treated under the name of the derivative equations. = There is one remarkable property of the limiting equation: viz. that when fi (p) O, then ƒ (p) has a greater or less value than it has when fi (p) is either a positive or negative quantity a little different from 0; or, in more technical language, (though belonging to a more advanced subject of study,) ƒ (p) is a maximum or a minimum. .... THEOREM XIII. If one of the roots r, of an equation Ax" + Ba"-1 + Lx2+ Mx + N = 0, be very small in comparison with all the others, we shall have, nearly, r1 = N .. rn For, let r2, 3, ....r, be the other roots; then (theor. VI.) we have M= {r1 (r2r,. • • • T n−1 + P3 P4 • • • • r n + r1 r s• • • • r n r 2 + . . . ) + r2 rg r • • • r } M=± {r1(r2r3 • • • r 3r 4 N=23 .rn. r3 Now since r, is very small in comparison with the other roots, the vinculated term which contains it as a factor in M is small in comparison with the term T2 Tз.... ̃, which does not contain it. Neglecting, therefore, this term, we have N Frir2r3. În N == r1, or r1 = ....Tn Μ' Scholium. This theorem enables us, after we have obtained a first distinct approximation, to obtain a closer one by mere division; and thence to still further reduce the roots of the equation, and especially that to which in any case we may be approximating. PROBLEM VII. To find the limits between which are situated the roots of any given equation, Ax2 + Bæ1 + . . . + Lux2 + Mx + N = 0. : 1. Find a reducing number k which will render the signs of all the coefficients positive for then the roots of the transformed equation will be negative, (Theor. VIII.) and hence k is greater than the greatest root of the equation, (Cor. 1. Prob. III.) or it is the superior limit of the positive roots. 2. Find similarly a number which will render all the coefficients of the transformed reciprocal equation positive; the reciprocal of this number will be less than the least positive root, or will be an inferior limit of the positive roots. 3. Change the alternate signs, and find the superior and inferior limits of the positive roots of this transformed equation: these limits, taken with negative signs, will be the limits between which all the negative roots lie. 4. To find how many roots lie within any given limits, a and b, of which a is the greater, reduce the roots of the given equation by the less of those numbers b; then, again, reduce the roots of this transformed equation by a-b. The difference of the number of variations of sign in these two transformed equations indicates the number of positive roots in the interval. In a similar manner, after changing the signs of the alternate terms, and of the two negative limits, we may find the numbers of variations in each reduced equation; the difference of which will be the number of negative roots in that interval. The first part of this rule becomes evident from Theor. VIII. and Cor. 1. Prob. III. and the latter, from combining it with Prob. III. itself. Scholium 1. As a practical course of procedure, it will be advisable to reduce by 1, 10, 100, ... rather than by intermediate numbers to these, till the utmost limit is obtained; and then to work with such intermediate numbers, as may be thought, from the state of the coefficients, most likely to make a small number of changes in the state of the coefficients as to order and number of signs. Proceeding with these till we have obtained two limiting consecutive numbers for one or more roots, the object of this problem will be attained. Scholium 2. When by narrowing the intervals of the substituted numbers, we find more than one variation continually disappearing in each of the substitutions made; these roots may be equal, or they may have minute differences, or any even number of them may be imaginary. If there be equal roots, the process of Problem VI. will find them. The only question, then, is to find whether an equation known to have only unequal roots, has any number of them imaginary, and how many; the remaining ones, of course, being real, and having differences less than that of the limiting substituted numbers between which they are indicated. Several methods of solving this problem have been proposed; but we shall here give only three of these criterions, those of De Gua, Budan, and Sturm; though those of Lagrange and Fourier well deserve to be studied by every one whose time and inclination lead him to pursue the subject further. See Lagrange, Resolution des Equations Numériques, p. 6, and Fourier, Analyse des Equations Determinées, p. 87. THEOREM XIV. De Gua's Criterion of Imaginary Roots *. This criterion is generally stated incorrectly. It should also be expressed more in detail than is usually done. Before stating it, however, it will be desirable to enter upon the examination of the principles from which it flows. 1. It is very clear from the reasoning in theor. VIII. that the rule of Harriot is true for all cases in which the roots of an equation are real, or constituted merely by the signs or prefixed to a real number, either integer, fractional, + or irrational. 2. It follows, then, that all cases in which this assumption being made leads to contradictory results, indicate, according to the number of those contradictions, so many of the roots not being constituted as above expressed; that is, so many of the roots are imaginary. 3. When we have a cipher-coefficient, such as Ox", it is either + Ox" or —0xTM ; the values of the expression in which it occurs being precisely the same in both cases. 4. The greatest number of negative roots in an equation which contains ciphercoefficients between any two actual coefficients, will be when all the ciphers are written with the same signs as one of the terms which form the extremes between which the ciphers are situated. 5. The least number of negative roots under the same circumstances will be *This property of cipher coefficients was first given by the Abbé de Gua, in the Memoirs of the French Academy, for 1743. All writers, after the original author, have, however, committed an oversight in estimating the number of conditions implied in this theorem, which they n(n+1) uniformly assert to be for n cipher coefficients. The conditions may, indeed, appear under different simultaneous forms: but their number cannot be more than as stated in the text above. 2 when the ciphers are taken with alternate signs, the first cipher being taken with the contrary sign to its adjacent actual coefficient. 6. The difference between the greatest and least number of negative roots indicated by taking the ciphers as in (4), (5), is the number of imaginary roots indicated by the sequences of the cipher-coefficients in the equation. 7. If cipher-coefficients occur in any of the transformed equations, the same rules apply, since no kind of transformation by real numbers can eliminate the imaginary part of the root. It is, however, to be carefully kept in view, that no proof is offered of the imaginary roots, indicated by the transformed equation, being different from, or the same with, those indicated by the given equation. The statement, then, of De Gua's rule will be as follows: 1 cipher-coefficients inter I. If between terms having like signs, 2n or 2n vene, there will be 2n imaginary roots indicated thereby. II. If between terms having different signs 2n + 1 or 2n cipher-coefficients intervene, there will be 2n imaginary roots indicated thereby. (1) Let there be 2n ciphers between like signs; then writing them as expressed in (7) we have Whence since only one root is negative in the latter case, and 2n + 1 roots in the former, there is a contradiction in the sign of 2n roots; which are, therefore, imaginary. Had the signs of the extreme terms, h, k, been both — instead of +, the series would have taken the form And the same contradiction with respect to 2n of these roots would have resulted. (2) Next, let there be 2n 1 ciphers between like signs: then +h+0+0+0+ +0+0+k, giving 2n negative roots. +h .... 0 + + 0 − 0 + k, giving 0 negative root. .... Hence there are, as before, 2n imaginary roots. (3) Let there be 2n ciphers between unlike signs: then (4) Let there be 2n + 1 ciphers between unlike signs: then +h+0+0+0+ .... +0+0+0− k, giving 2n + 1 negative roots. 0 + − 0 + 0 − 0 — k, giving 1 negative root. .... Hence, again, there are 2n imaginary. The theorem of De Gua is, therefore, established universally, and the statement given in its most general form *. It has been well remarked by Mr. Horner, (Math. Repos. vol. v. p. 27,) that though the direct application of this "criterion can only occur incidentally, yet its application is capable of an extension beyond what is at once apparent. To cite an example; when the mth coefficient changes its sign in passing from one set to another, while those which immediately precede and follow it are and continue to be identical, the existence of zero between like signs somewhere in VOL. I. For example, x3 + x + 1 = 0 having the coefficients 1 + 0 + 1 + 1, has two imaginary roots; and the equation 3 + 5 = 0 having the coefficients 1±0 ±0 ± 5, has also two imaginary roots. EXAMPLES FOR PRACTICE. 1. How many imaginary roots are there in the equations a 6x2 + x = 0, 2810, and x1. 6x2+2x+12= 0. 2. The equation æ 5x1 + 20x3 + x = 100 has one pair of imaginary roots indicated by its present state, and reducing the roots by a certain number to be found, will show another pair. What is that reducing number, and which are the places at which the imaginary roots are indicated? 3. Has the equation x1 — 4x3 + 8x2 — 16x + 20 = 0 any imaginary roots? THEOREM XV. Budan's Criterion *, The theorem may be stated thus:— as arranged by Horner. If in transforming an equation by any number r, there be n variations lost, and if in transforming the reciprocal equation by there be m variations left; then there will be at least n m imaginary roots in the interval 0, r. r (1) There can be no root of an equation infinitely great except the absolute term be also infinite. The roots of equations, such as generally occur, are, therefore, finite. (2) The reduction of the roots of an equation by or infinity, must therefore render all the roots negative; and hence give only permanencies of sign in the transformed equation. (3) Imaginary roots enter an equation in pairs, as has been shown in Theor. V. the interval may be suspected. Should all the previous signs in these sets be alike, the proba bility is increased." He might, indeed, have spoken even more decidedly; as by other means we can show that it amounts to all but absolute certainty. In successive transformations this remark will often be of considerable utility to be borne in mind. * This criterion was first given by Budan in his Nouvelle Méthode pour la Résolution des Equations Numériques, p. 36; but the form in which it now appears is due to Mr. Horner, and in any other it is next to useless. It would almost appear from a note to Lagrange's Traité de Résolution des Eq. Num. (p. 169,) that he did not seize its import: at all events, he formed an inadequate notion of it, and raised an objection to it which is altogether invalid. The following observations upon using it will be useful to the student :— 1 r 1. It will always in practice be most convenient to take the transforming interval r equal to 1; as then the reciprocal is also equal to 1. When the interval is a prime number different from 2 or 5, it often becomes troublesome to reduce the fractional remainders; and it is not often safe to neglect them, as the whole force of the criterion may be destroyed by a very small quantity. Besides, though imaginary roots can never be indicated by this criterion except they exist, they may exist without being indicated by a specified interval. The smaller the interval, therefore, the greater the probability of their detection. Hence, except in rare cases, it is better to take the interval 1. 2. When no indication is supplied by the interval 1, it will be most convenient to use |