are comprehended between any two numbers, a, ß, («<ß), substitute a in the series (x); write the signs of all the results, and count the number of variations; next, substitute ẞ in the series (x), write the signs of all the results, and count the number of variations. As many variations as the second set of signs has fewer than the first, so many real roots of the equation X=0 are comprehended between a and 3. In dividing X by X, let Q, be the quotient; then, since X,, is the remainder with all the signs changed, Hence, 1st, two consecutive functions of the series (x) cannot become zero for the same value of z. For if, for the same value of x, X=0 and X=0, the equation X--XQ-X+1, which is comprehended in the series (x), becomes 0=0—X2+1• Therefore X+=0; and, proceeding from equation to equation, it would be proved that X=0, which is contrary to the hypothesis. 2d. If an intermediate function of the series (x) becomes zero for a certain value of x, the values of the preceding and following function are of contrary signs. For when X,, for example, becomes zero, X-1=X„Q»−X»+1 becomes X-−Xn+1• a. Let it be supposed that the value of x is made to begin with a, and to increase in a continuous manner, and that these continuous values are introduced into the series (x); so long as x has not reached a value which destroys one or more functions of the series (x), it is evident that the signs of none of these functions are changed, and, consequently, that the series presents always the same succession of signs. When x has attained the value a, for which one or more of the intermediate functions X, X, · becomes zero; then if X, is one of these functions, the functions X-, X+1, which precede and follow X,, ought for this value to be different from zero, and to have contrary signs; so that, giving to the vanishing function either the sign + or, the series of the three functions X, X, X+1 will present always one variation, and only one. Since neither of the functions X,-1, X+1, changes sign between x=a and x-a, they must, for x=a, be of contrary signs; therefore, whatever was the sign of X, before the substitution of a for x, the series of these three functions must have presented one variation, and only one. b. When a becomes a+h (h being positive, and so small that no root of the equations X=0, X,+1=0 falls betweens a and a+h,) in this interval neither of the functions X-, X+1 can change its sign. They must, therefore, have contrary signs. Consequently, whatever the sign of X,, there is still only one variation in the series of the three functions. If any other intermediate function become 0 for x=a, it must in like manner be situated between two functions which do not become zero, and the preceding considerations must apply to these three functions. Therefore, after the substitution of the value a for x, there are in the series (x) the same number of variations as before, although they may be differently distributed. c. The succession of signs which is obtained by substituting for x values a little greater than a remains unchanged until x is increased so as to exceed the value which makes some function of the series (r) to disappear. But if this value does not reduce the first function, X, to zero, it follows from the preceding part of this demonstration that the variations of the series (x) are the same in number after the substitution of this value as before it, and that this number of variations cannot change unless x is increased so as to exceed a root of the equation X=0. It is necessary, therefore, to compare the signs which then belong to the series (x) with those which it has for values of x, a little less than a. d. Let a be a root of the equation X=0; make x=a+h (h being an indefinitely small positive quantity); denote by A, A', A"... the values of the polynomial X and the derived polynomials for x=a, and by II the value of X corresponding to a+h. Then H=A+A′h+1A′′h2+, &c. But since a is a root of X, A=0; therefore H-A'h+A"h2+, &c., in which expression A' cannot be zero, because the equation X=0 has no equal roots. Making h a common factor of the terms of the second member of the last equation, H=h(A'+A′′h+, &c.), the value of h being indefinitely small and positive, the sign of the whole quantity A+A"h+, &c. is the same as the sign of the term A'; therefore the sign of H must be the same as the sign of A'. To discover what was the sign of X before z reached the value a it is necessary to make x=a-h. The result, H', of this substitution is obtained by changing hinto -h in the expression of H; wherefore II-h(A'—A′′h+, &c.) From which it is evident that for very small values of h the quantity H' has a sign contrary to that of A'. Therefore, when the value of x is less than a the series (r) has a variation from X to X,; and when the value of x exceeds a this variation is replaced by a permanence. But for values of x a little less than a or a little greater than a, the remaining part of the series (a) ought always to have the same number of variations, even although the value r=a should reduce some intermediate function to zero. Therefore when x, by increasing, comes to exceed a root of the equation X=0, the series (x) loses one variation. Continuing to augment x, the remaining number of variations of the series (x) is not changed until x comes again to exceed a root; then another variation is lost. e. The same thing happens for every instance in which x exceeds a new root; so that the number of variations lost by the series (x) is always equal to the number of roots of the equation X=0 comprehended between the value x=a, with which the substitutions are commenced, and the value x=3, with which they are concluded. 280. In the successive divisions made to find X,,, X,,,, &c. the dividends and divisors may be multiplied by any positive numbers; for by such multiplication none of the signs are changed. a. If in the series (r) a function X is found which does not become zero for any value between x=a and x=3, and if it is required to discover the number of roots comprehended between a and ß, the series may be made to end with the function X. Then making a to vary from a to 3, the principles which are applicable to complete series may be applied to the partial series X, X, X, X .. XX. b. If either of the limits e, ẞ reduces an intermediate function to zero, this function is always situated between two others which do not disappear, and which are of contrary signs; so that the function which becomes zero may be taken, indifferently, with the sign + or the sign, or omitted altogether. But if for either of the limits, 6 for example, X vanishes, it is to be concluded, first, that ẞ is a root. Next it follows, from Article 279 d, that if values a little greater than ẞ are given to x, there is a permanence from X to X,, whilst in the remaining part of the series there is the same number of variations as for x=. Consequently the general rule 279 e. gives the number of real roots comprehended between a and a quantity a little greater than 3 281. If X, is not the derived polynomial of X, but a polynomial which has no factor in common with X, and which takes a sign contrary to X for values of x which differ very little in defect from any root whatever of X=0, and if this polynomial X, is employed to find X,, X,,,, &c. in the same manner ዘለ as the derived function of X is employed in Article 279, the theorem of Sturm subsists equally with the new series X, X, XX &c. For if the details of the demonstration are resumed, it will be found that the new intermediate functions have the same properties; likewise in passing from a value of x a little less than a root of X=0 to a value of x a little greater than a root of X=0, the function X must change its sign, whilst the function X,, which has no factors in common with X, must preserve the same sign. 282. If the equation X=0 contains equal roots; if, for example, X= (x-a)" (x—b)"'(x—c)(x−d) . . . . the divisions made to deduce from the polynomial X and its derived polynomial X, the functions X,,, X, &c. must lead to an exact divisor X,, a function of r which is the greatest common measure of X, X,; and from the composition of this common measure (Art. 251) if ¡X, X, are divided by X, the quotients are X = (n(x-b)(x-c)(x-d)...+n'(x—a)(x—c)(x—d) ... { +(x−a)(x−b)(x−d) + &c. ·+(x−a)(x−b)(x−c) of which V contains no equal factors, and V, has no factor in common with V. Let x-a be a real factor of X, and make H=(x-b)(x-c)(x−d). ... +(x−b)(x−d) . . . +, &c., Therefore V=(x-a)H; V=nH+(x-a)H. Now, from the nature of the operations which serve to discover the greatest common measure of X, X, and to find the quotients V, V, these quotients, and by consequence also H, H, ought not to have imaginary coefficients, since there are none in either X or X,; whence, for values of a very little less than a, the factor x-a being very small, V, will have the sign of H, which sign is evidently contrary to that of V. Therefore, in applying the theorem of Sturm to the equation V=0, the quotient V, may be employed instead of the derived function of V. by X,. Likewise the functions V,,V,,, which it is necessary to deduce from V, V, by the rule (Art. 236), are the quotients of X,,, X, Therefore the two series X, X, X, V, V, V Xr .... differ from each other only by a common factor; and whatever the sign of this factor, it is evident that the two ought to present the same variations when a value of x which does not destroy X is substituted in them. Therefore, by the application of the theorem to the first series, it can be determined how many roots of the equation V=0 there are between two numbers, a and B, neither of which renders X=0; or (which is the same thing) how many real roots there are in the equation X=0, abstraction being made of the degree of multiplicity. Hence, in the equation X=0, the number of real roots comprehended between a and 3 is equal to the excess of the number of variations in the signs of the functions X, X, X,,. X, for x= above the number of variations for x=3, abstraction being made of the multiplicity of the roots. 283. This theorem affords a simple means of establishing the conditions of the reality of the roots of an equation. Since the real roots are all comprehended between two limits -L' and +L, the one negative and the other positive (which may be chosen = ∞), the question is reduced to the investigation of the conditions necessary in order that from x=-L' to x=+L the series X, X, X, X, may lose a number of variations equal to the degree of the equation. Supposing m the degree of the equation, the series (2) must lose m variations. To lose m variations it must contain at least m+1 terms; and since it cannot have more, the functions X, X, X,,, . . . are m+1 in number. They are consequently of the respective degrees m, m−1, m−2, &c., the last which does not contain r being X When in the polynomials in a very large numbers, positive or negative, are substituted for x, the results are of the same sign as if each polynomial were reduced to its first term (Art. 258); whence, in this investigation, attention need be given to the first term only of each of the functions X, X, X,,, &c. Taking the equation X=0 in the ordinary form, x+Px"1+QxTM-2+&c.=0, the first term of X is x, that of X, is mr-, and those of X,,, Xu • are certain functions composed of the coefficients P, Q, &c., which are determined by successive divisions. Representing these functions by G,,, G., • • and arranging the m+1 quantities in order, thus x", mx"-1, G, xTM-2, G-3 G the question is reduced to the investigation of the conditions which cause this series to lose m variations in passing from x=-L' to x=+L. Now, that this may be possible, it is necessary that there be m variations after the substitution of -L', and m permanences after the substitution of +L. On the other hand, in this series the powers of a diminish by 1 from term to term; consequently, if it has permanences only when x=+L it must have variations only when r--L'. Whence the conditions sought are reduced to those which are necessary in order that the series may have positive coefficients only; that is to say, to the conditions that G,,>0, G,,,>0 . . . ., G„>0. -1; It is evident that these conditions cannot be more in number than m― but they may be fewer, for some of the inequalities G,,>0, G,,,>0... may be reducible to each other. As an application, let it be required to find the conditions necessary for the reality of the roots of the equation, x3+Qx+R=0. In this example m=3 and m-1=2. The number of conditions is therefore 2; these are, that G>0 and G>0. To obtain G,, and G,,, it is necessary to find X,, and X which is accomplished as follows: X=x3+Qx+R.·. X,=3x2+Q 3 3x2+Q)3x+3Qx+3R(x 2Qx+3R ..-2Qx-3R=X,,. -2Qx-3R)12Q2x2+4Q3(−6Qx+9R -18QRx+ 4Q3 4Q3 + 27R2 ... —4Q3-27R2=X,,, To avoid fractional expressions the dividend in the first division is multiplied by 3, and in the second by 40; these factors being positive the signs of X,,, X,,, are not affected by the multiplication. It is thus found that X,,=-2Qr-3R, and that X,,,=-4Q3-27Ro. Whence the inequalities G,,>0, G,,,>0, become -2Q>0, -4Q3—27R2>0, and changing the signs, 2Q<0, 4Q3+27R2<0. The first of these conditions is comprehended in the second; for the term 27R being necessarily positive, the second condition requires that Q be negative. The only condition absolutely necessary is therefore 4Q$+27R<0. When this is fulfilled the roots of the cubic equation are all real; when it is not, one root is real and two are imaginary. 284. In order that the roots of the equation X=0 may be all real, it is necessary that the signs of the series (x) present only variations for x=— ∞ and permanences for x=+∞. Let it be supposed (as in Art. 283) that the series (x) contains m+1 terms, which diminish in degree regularly by 1; then, since any two consecutive terms of the series are, the one of an even, the other of an odd degree, if these two functions have the same sign for x=+, they must have contrary signs for a= c; and if they have the same sign for x=-∞, they must have contrary signs for x=+∞. If therefore the signs of the functions X, X, X, . or of the first terms of these functions, viz. x”, mx”—', G ̧x-2, Xm for x= ∞, and for x=+ ∞, are written the one series under the other, each variation in either series will correspond to a permanence in the other series. Therefore the number of permanences for x= ∞ is equal to the number of variations for a=+ ∞. Let n=the number of permanences for x=— ∞. ... n=the number of variations for r=+∞c. Whence, the whole number of signs being m+1, the number of permanences for r=+ ∞ is m−n; and the number of variations lost between x=− ∞ and x=+ cc is m—n−n, or m-2n. Consequently, if the number of permanences for x=— ∞ is n, the number of imaginary roots of the equation is expressed by 2n, and the number of real roots by m-2n. USE OF STURM'S THEOREM. 285. Let the equation which is to be resolved contain no equal roots, and let the series (a) be formed; it is evident that by the substitution of 0 for z each of the functions X, X, X,,, &c. is reduced to its last term, and that by the substitution of values above a certain positive limit (+L) each function is made to take the same sign as its first term. Therefore by inspection of the series (r) the number of variations of the series (0) and also the number of variations of the series (+L) are known. The difference between these two numbers indicates how many positive roots are contained in the equation. The number of negative roots is obtained with equal facility, by observing that beyond a certain limit,—L', all negative numbers impart to each function of the series (r) the same sign as to its first term. For convenience +x and ∞ are taken instead of +L and -L'; this being an abridged manner of denoting two numbers which may be as great as we please. As the negative roots are rendered positive by changing x into -r, it is necessary to treat of the positive roots only. The separation of these is effected by substituting in the series (x) increasing positive numbers, 0, a, ß, 7, &c.; then by counting the number of variations lost at each substitution, the number of roots comprehended between 0 and a, a and 3, &c. will be obtained. These substitutions ought to be continued until a series is found whose variations are the same in number as those of the series (+ x). Farther it is needless to proceed, for there can be no root above the last number substituted. Admitting that there are several roots between a and B, one or more intermediate substitutions must be made, then the number of variations lost by |