these positive factors; and substituting for a the proposed series of values, b1, b2, bз, &c., we have these results: (b1-a) (b-a) (b,—a ̧) (b,—a1).. • = +.+.+.+. .... = ·+.+. = + = + 1 + Cor. 1. If two numbers be successively substituted for x in any equation, and give results with different signs, then between these numbers there must be one, three, five, or some odd number of roots. Cor. 2. If the results of the substitution in Cor. 1 are affected with like signs, then between these numbers there must be two, four, or some even number of roots, or no root between these numbers. Cor. 3. If any quantity q, and every quantity greater than q, renders the result positive, then q is greater than the greatest root of the equation. and Cor. 4. Hence, if the signs of the alternate terms be changed, and if p, every quantity greater than p, renders the result positive, then p is less than the least root. EXAMPLE. Find the initial figure in one of the roots of the equation x3-4x2-6x+8=0. Here one value of x does not differ greatly from unity, for the value of the given polynomial, when a=1, is -1, and when x=9, it is found thus: 1-4-6 +8 (+9 •9-2.79-7 911 −3·1—8·79+·089 ... V=+089. Hence the former value being negative, and the latter positive, the initial figure of one root is ⚫9. PROPOSITION VII. Given an equation of the nth degree, to determine another of the (n−1)th degree, such that the real roots of the former shall be limits to those of the latter. Let a, a, aз, α19 -1 a be the roots taken in order of the equation 2+A+A112"-2+ A2-1x+A=0; then diminishing the roots of this equation by r (Prop. I, p. 280), we have the following process, viz.: Whence C=An-1+rB2-2+rCn-2 =A„-1+ 7(An-2+ r B„-3)+7(An~2+7°Вn~3+7°Сn~3) Or, C1-1=nr1+(n−1) A ̧r1~2+(n−2) A11212¬3+ . . .2A„-27 + A ̧-1 • . (1) Again, the roots of the transformed equation will evidently be a1—r, a2-r, A3—r, ɑ4—7', ・ ・ ・ ・ a-r, and as we have found the coefficient, C-1, of the last term but one, in the transformed equation, by one process, we shall now find the same coefficient, C-1, by another process (Prop. IV, p. 277;) hence we have C1-= (r—a1) (r—a2) (r—α3) Now, these two expressions which we have obtained for C-1 are equal to one another, and therefore whatever changes arise by substitution in the one, the same changes will be produced, by a like substitution, in the other; hence, substituting a, a,, a3, &c., successively for r in the second member of equation (2), we have these results: But when a series of quantities, a1, ɑ2, ɑз, α, &c., are substituted for the unknown quantity in any equation, and give results which are alternately + and, then, by Prop. VI, these quantities taken in order, are situated in the successive intervals of the real roots of the proposed equation; hence, making C=0, and changing r into x, we have from equation (1) n-3 nx11+(n−1)A ̧ï12+(n−2) A1113+.... 2A ̧-2+A-1=0... (3) an equation whose roots are therefore limits to those of the original equation, x"+A1x"1+A112"-2 + . . . . A„-1~+A„=0, and the manner of deriving it from the proposed equation is evident. Let a1, A2, A3, A4, &c., be the roots of the proposed equation, and b1, b2, b3, &c., those of the derived equation (3), ranged in the order of magnitude; then the roots of both the given and the derived equation will be represented in order of magnitude by the following arrangement, viz.: a1, b1, ɑ2, b2, ɑз, b3, ɑ4, b4, ɑ5, b5, &c. . . Cor. 1. If a, a,, then r-a1, will be found as a factor in each of the groups of factors in equation (2), which has been shown to be the limiting equation (3), and therefore the limiting equation, and the original equation, will obviously have a common measure of the form x-a1. Cor. 2. If a a2=a1, then (r-a1) (r—a,) will occur as a common factor in each group of factors in (2); that is, the limiting equation (3) is divisible by (x-a1)2; and therefore the proposed equation and the limiting equation have a common measure of the form (x—a1)3. Cor. 3. If the proposed equation, have also a;=a, then it will have a common measure with the limiting equation of the form (x-a1)2 (x—α), and so on. Scholium. When therefore we wish to ascertain whether a proposed equation has equal roots, we must first find the limiting equation, and then find the greatest common measure of the polynomials in the first members of these two equations. If the greatest common measure be of the form then the proposed equation will have (p+1) roots=a1, (q+1) roots=a2, (r+1} roots=a3, &c. The equation may then be depressed to another of lower dimensions. BUDAN'S CRITERION For determining the number of imaginary roots in any equation. .... ... 172. If the real positive roots of an equation, taken in the order of their magnitudes, be a1, а2, aз, as a, where a is the smallest, and if we diminish the roots of the equation by a number h greater than a1, but less than a2, then the roots will be a,-h, a2-h, a,—h, a-h, and the first of these will now be negative. But the number of positive roots is exactly equal to the number of variations of sign in the terms of the equation, when the roots are all real; and as we have changed one positive root into a negative one, the transformed equation must have one variation less than the proposed equation. Again, by reducing all the roots by k, a number greater than a2, but less than aз, we shall have two negative roots, a1-k, a,―k, in the transformed equation, and therefore we shall have two variations of sign less than in the proposed equation; for two positive roots have been reduced so as to become negative ones. Hence it is obvious, that if we reduce the roots by a number greater than all the positive roots will become negative, and the transformed equation, having all its roots negative, will have the signs of all its terms positive (Prop. IV. p. 277), and all the variations have entirely disappeared. an We see, then, that if the roots of an equation be reduced until the signs of all the terms of the transformed equation be +, we have employed a greater number than the greatest positive root of that equation; and therefore its reciprocal must be less than the smallest real root of the reciprocal equation. Now, if we take the reciprocal equation, and reduce its roots by the reciprocal of the former number, we should have as many positive roots left in this transformed reciprocal equation as there were positive roots in the proposed equation, unless the equation has imaginary roots; hence the number of variations lost in the former case should be exactly equal to the number left in the latter, when the roots are all real; and, consequently, if this condition be not fulfilled, the difference of these numbers indicates the number of imaginary roots. To explain this reasoning more clearly, we shall suppose that an equation has three positive roots; as, for instance, 1,2.5, and 3. Now, if the roots of the proposed equation be reduced by 4, a number greater than 3, the greatest positive root, the three positive roots in the original equation will evidently be changed into three negative ones in the transformed one, and hence three variations must be lost. Again; the equation whose roots are the reciprocals of the proposed equation, must have three positive roots, 1, 4, and ; and it is evident that if we reduce the roots of the reciprocal equation by 4, the reciprocal of the former reducing number 4, we shall not change the character of the three positive roots, because is less than the least of them, and 1—4, 4—4, —, are all positive; hence the three variations introduced by the three positive roots must still be found in the transformed reciprocal equation, and therefore three variations are left in the latter transformation, indicating no imaginary roots. The theorem may, therefore, be stated thus: If, in transforming an equation by any number r, there be n variations lost, and if in transforming the reciprocal equation by (the reciprocal of r,) there be m variations left, then there will be at least n―m imaginary roots in the interval 0, r. For there are as many positive roots in the interval 0, r, of the direct equation, as there are between and of the reciprocal equation; hence, if n, the number of variations lost in the transformation of the direct equation by r, be greater than m, the number of variations left in the transformation of the reciprocal equation by, there will be a contradiction with respect to the character of a number of the roots, equal to the difference n―m. Hence these roots are imaginary. Here two variations are lost in the transformation of the direct equation, and no variations are left in the transformation of the reciprocal equation; therefore, this equation has at least two imaginary roots; and it has only two; for the sign of the absolute term is negative, implying the existence of two real roots; the one positive, and the other negative. DEGUA'S CRITERION. 173. In any equation, if we have a cipher-coefficient, or term wanting, and if the cipher-coefficient be situated between two terms having the same sign, there will be two imaginary roots in that equation. In the former of these we find two permanencies and five variations, and in the latter we have four permanencies and only three variations; hence, if the roots are all real, we must, in the former case, have five positive and two negative roots, and in the latter, three positive and four negative roots (Prop. VII. p. 279); hence we have two roots, both positive and negative, at the same time, and therefore these two roots cannot be real roots. These two roots, which involve the absurdity of being both positive and negative at the same time, must therefore be imaginary roots. In nearly the same manner it may be shown that (1.) If between terms having like signs, 2n or 2n-1 cipher-coefficients intervene, there will be 2n imaginary roots indicated thereby. (2.) If between terms having different signs, 2n+1, or 2n cipher-coefficients intervene, there will be 2n imaginary roots indicated thereby. Ex. The equation x1—x3+6x2+24=0 has two imaginary roots; for the absent term is preceded and succeeded by terms having like signs, and the equation 2+1 having the coefficients 1+0+0+1 has also two imaginary roots. EXAMPLES FOR PRACTICE. (1.) How many imaginary roots are in the equation x5+x3-2x2+2x-1=0? (2.) Has the equation x1—2x2+6x+10=0 any imaginary roots? 174. The most satisfactory and unfailing criterion for the determination of the number of imaginary roots in any equation is furnished by the admirable theorem of Sturm; which gives the precise number of real roots, and consèquently the exact number of imaginary ones; since both the real and imaginary roots are together equal to the number denoted by the degree of the proposed equation. PROPOSITION VIII. To find the number of real and imaginary roots in any proposed equation. The acknowledged difficulty which has hitherto been experienced in the important problem of the separation of the real and imaginary roots of any proposed equation, is now completely removed by the recent valuable researches of the celebrated M. STURM; and we shall now explain the theorem by which this desirable object has been so fully accomplished. has no equal roots, and let X1n Ax+(n−1)Bx" be the derived function, arising by multiplying each term of the equation +H |