1 (4) There are as many positive roots in the interval 0, r, of the direct equation, as there are between and of the reciprocal equation. For the roots of the reciprocal equation are the reciprocals of the roots of the direct equation; and hence must lie between the reciprocals of those limits of the direct equation. (5) If then the number of variations n, lost in the direct equation by passing over the interval r, be greater than the number left m in the reciprocal equation, after transformation by, there will be a contradiction with respect to the character of a number of them, equal to the difference n — m. These roots, therefore, are imaginary. To take one example, find whether the equation æ3 31x = 101 has imaginary roots. Reduce by 1; then the work will stand thus: 8x+11x3 39x2+ 2.... where four variations are already lost, 5, which narrows the limit still more, as we have the intervals 0, ·5 and ·5, 1; and the reciprocal of 5 being 2, we have an easy working number for the reciprocal equation, if both roots be still left doubtful. If there be one variation, and one only, lost, the roots are separated, one being between 0 and 5, and the other between 5 and 1. Should it still be uncertain, it will be convenient to reduce either the given equation or that in 5 by 2, since the reciprocal 5 is an easy number for the reciprocal transformation. 3. It will sometimes conduce to clearness of conception to multiply the roots by 10, 100, 1000, &c., in accordance with what is said at p. 213 of this work. This, however, is only the case in early practice of the method, since the process itself is not virtually altered by it. This step simply adds one cipher to the second coefficient, two to the third, three to the fourth, &c. 4. It will also be desirable in these successive transformations to keep De Gua's Criterion in view, both in its strict form, and with the modification suggested at the foot of page 225. 5. It will generally happen when we have taken too wide an interval that the ambiguity may be very simply removed as follows: Reduce the reduced reciprocal equation by unity at a time till the variations are lost in pairs, or the roots separated belonging to the doubtful interval. Apply the criterion to this reciprocal equation; and if the roots be indicated as imaginary, they are so; and then the roots of the original equation, which are functions of these, are also imaginary. To take an example, let x + x3 + 4x2 - 4x + 1 = 0 be given. In the direct transformation two variations are lost, and in the second two variations are left; hence no conclusion whether they are or are not imaginary can be drawn from this result. Take the reduced reciprocal equation, therefore, and we shall have In these two transformations we have now two imaginary roots by Budan's criterion; and hence the two corresponding positive roots of the given equation are imaginary. Let X = 0 be an equation of the nth degree, and let X1 = 0 be its first derivative; and let the given equation be free from equal roots. Perform the operation employed in finding the greatest common measure, always, however, changing the signs of the several divisor-remainders: and denote the series of functions which result, together with the given ones, by X, X1, X2, . . . Xm · X. Then if a and b be any two numbers substituted in these several functions, the difference of the numbers of variations of sign in the results of these substitutions expresses the number of real roots lying between a and b. 1. Denote the successive quotients by Q1, Q2, α .... Q-1: then Now as the degree of remainder is clear of x the function is diminished a unit at each stage, the final and as the equation X = 0 contains no equal roots, X, cannot be zero, or in other words, X, is a number. 2. No two consecutive functions can become zero for the same value of x. For if it be possible, let them beX-1 and X. Then since Xm_1 = Qm Xm — Xm+19 No two consecutive and Xm-1 = 0, and Xm =0 at the same time, we also have Xm+1 =0. But Xm = Qm+1 Xm+1 Xm+2, and hence, for the same reason, Xm+2 0; and similarly all which follow Xm become zero for the same value of x, amongst which is X„. Now it has already been shown in (1) that X, cannot become zero: and hence also X cannot become zero at the same time with X-1 functions can, therefore, vanish with the same value of x. 3. If any value of a cause one of the functions, as X,, to vanish, it gives to the two adjacent ones equal values with contrary signs. This is evident from the connecting equation X-1 = Qm Xm — Xm+1, which in this case becomes Xm-1 = Xm+1. 4. If such a value be given to x as shall make one of the intermediary functions X1, X, .... X-1, vanish, without making X = 0, there will be a change in the order of the signs produced at this stage of the variable values of x, but no change in the number of variations. m For let X be that in which any special value of x makes the result zero. Then, considering the three consecutive functions, Xm-1, X and Xm+1, we have seen that in such case X-1 and X+1 have contrary signs (3); and the series of signs will therefore be + X-1, +0,- X+1 or X-19 ± 0, +X+1. That is, writing only the signs, we have the following combinations: − 3 · · . . . . . (1) Now denoting by a, a root of X, = 0, it is established, (Theor. IX.) that if numbers greater and less than a be substituted in Xm, the results will have contrary signs; and hence, in passing through X = 0, the signs will undergo the changes indicated in (1) or (2): but so far as the number of variations is concerned, these are all precisely alike, and differ only in the order of their succession. No variation, therefore, is lost or gained in the passage of X, from a value greater than a to one less, or the converse. Moreover, that which holds true for X, holds true for any other function X+; and hence it holds true universally. No variation, therefore, can be lost or gained amongst the intermediate functions, though their order of succession should be changed in any way whatever. 5. Every time a value of a coincides with a real root r, of the equation X=0, one variation, and one only, is gained in a descending series of values, or lost in an ascending series. Since (4) no variation can be lost or gained amongst the intermediate series of functions X1, X2,.......... X, it follows that the only case in which a change in the number of variations arises, is by its occurring between X and X1. Now, by hypothesis ƒ (r) = 0; and hence, taking the quantities r — h and r + h to represent the substituted numbers less and greater respectively than r, in the equations X = 0, and X1 = 0, we get h3 .... 1.2.3 But in all these functions we may find a value of h so small, that the value of the whole series shall be the same with that of its first term. If, then, we take an ascending series of values for the substitutions, the signs before and after the passage of a through r will be — + and ++; that is, a variation will be converted into a permanence, or one variation is lost. It has also been shown, that no variation can be lost amongst the intermediate functions; and hence only one variation can be lost in passing through a root of X = 0, from less to greater values of x. Had we taken the descending series of values for a, the reverse would have taken place; viz. ++ converted into +, or one variation only be gained in the passage through r *. * During the passage in the values of a in the several functions from one root of X = 0 to another, we have seen that any difference of order of succession may occur, but no difference in the total number of variations: and it may conduce to the clearness of our conception how a variation may be introduced or lost in passing through the next lower or higher root of X = 0. We have seen (theor. IX.) that if r1, 72, ......, ranged in the order of magnitude, be the roots of X=0, and P1, P2, P-1, be those of X1 = 0, ranged similarly, then these several values ranged in the order of magnitude will be ...... ............ It hence follows, that in passing from r1 to r2, we pass through p, a root of X1 = 0; and 6. Let now a and b be any two numbers substituted in the series of functions, there will be one variation, and one only, gained or lost, according as we use ascending or descending values of x in passing from one value to the other, every time we pass through a real root of X = 0. There will hence be as many variations gained or lost as there are real roots between a and b, and no more than these can be gained or lost. There will be as many more variations in the series of signs of the functions arising from one substitution, than there are in the series of signs arising from the other, as there are real roots, neither more nor less. The difference in the number of variations in the two series of functions under these two substitutions, expresses, therefore, exactly the number of real roots in that interval. Sturm's criterion is, hence, fully established. Cor. 1. Applying this to find at once the entire number of imaginary roots of an equation, we have only to take a positive limit greater than the greatest positive root of an equation, and a negative limit numerically greater than the greatest negative root of the same equation: then the difference in the number of variations of the two series of results of the substitution of these limits in X, X1, X2, ..., X., will be the number of real roots of the equation, the remaining ones being imaginary. But as + and 1 0 1 though limits too wide for approximation, are sufficiently near for the present purpose: and as in this, like the more restricted limits above spoken of, the signs of the entire functions will be the same with those of their first terms, we may at once obtain the number of real roots, which is the difference of the number of variations in the true series. For the particular character of the roots in any given interval, it will, however, be necessary to form the values of the functions for the two numbers which form the limits. Scholium. It is very obvious, that except the changed signs, Sturm's functions are precisely those which occur in seeking for equal roots; and hence, if such occur, they will be made apparent, and the given equation depressed by these roots may be resumed as an original equation, which, from its being of lower dimensions, will create less difficulty in finding the functions *. Let us take as an example the following equation: Given 25+ 4x4 ·2x310x22x962 = 0. hence the signs of X1, when quantities less and greater than p, are substituted for a, must be changed, till we arrive at p2. But before arriving at p2 we pass through r2, and hence in this interval X = 0. From the value, therefore, ever so little above r1, to the value ever so little below P19 X and X, retain their signs unchanged; but at this point, p1, the sign of X1, changes to the opposite, and continues to retain it till we arrive at p2, when it again changes. The series, therefore, will be, supposing r, the greatest root of X = 0, greater between between between between between than r1r and 1 x + Ρι and r2 r2 and P2 P2 and r3r3 and p3 X1+ + + And so on, as the two last are a repetition of the first two signs, and the same order will continue to the end. * Sturm investigates a modification of his method, which dispenses with the resumption of the process with respect to the depressed equation. Nothing in point of labour is, however, saved by it. Hence, the difference of the number of variations of sign being 3 the equation has but one real root. However, it will be necessary to narrow the limits between which the substitutions are made in order to effect the entire solution. If x=0 we have the signs ·++ − +, or three variations; and hence there is no negative real root: but by Harriot's law of signs, there are two of the roots negative, there being two permanencies of signs in the given equation. These two negative roots are therefore imaginary. Again, take 3 and 4 for the values of x in the functions: then, when x = 3 the signs are +++ − +, and there are no variations lost in the series of signs; hence there is no real root between 0 and 3. When = 4, the signs are ++++ lost; hence there is one real root between 3 and 4. Moreover, the signs undergoing no change for any greater value of x, there is no real root greater than 4; or the only real root lies between 3 and 4. We shall resume this example presently for the purpose of completion of the entire process of solution 2. Given 23 real or not in each. 3. In a1 2x3 - 7x2+10x+10= 0, all the roots are real. 4. In 25+ 2x4 13x3 3x2 9x=19, there is one real root, two equal roots, and two imaginary roots. * It may be useful to the student to compare the determination of the character of the roots by Budan's criterion with that in the text by Sturm's method. (1) Reduce the roots by 1; then we have +++++, or 2 lost. (2) Reduce the reciprocal by 1; then we have Hence there are two imaginary roots in the interval 0, 1. or 0 left. (3) Reduce the roots of (1) by 2: then the signs are +++++, or 0 lost. Hence there is one real root between 3 and 4. To find the negative roots, change the alternate signs: then (5) Reduce by 3; and we have the signs +++++, or 0 lost. (6) Reduce (5) by 1; and we get ++++++, or 2 lost. (7) Reduce the reciprocal of (5) by 1; then + + + + + +, or 0 left. There are hence two negative roots between 3 and 4, and they are imaginary. |