To find the negative root, we have the following operation: 24 22112 126336.. 1816167 128152167 1827561 1299797 28.. 184 012707 130163740707 184127241 -100 (3.43357786336599 54 -46.... 416896 -43104.... 384456501 -46583499... 390491222121 -75343767879 65189289046 -10154478833 9128951421 -1025527412 912928254 -112599158 104335040 For the positive root we have a similar operation, 1 +1 +1 +3 -100 (2·8028512181582; but this we shall leave for the student to perform, and the two roots will be found to be (4) Find the roots of the equation x+2x1+3x3+4.x2+5.x—20=0. Here we have V = x2+ 2x1+ 3x3+4x2+5x-20 Hence the difference of variations of sign indicates the existence of one real and four imaginary roots, the real root being situated between 1 and 2. alw Hence the real root is nearly 1∙125790; and by using another period of ciphers we should have the root correct to ten places of decimals, with very little additional labor. ADDITIONAL EXAMPLES FOR PRACTICE. (1) Find all the roots of the equation x3-3x-1=0. +x2-500=0. (4) Find the roots of the equation x3+x2+x-100=0. x+4x3-3x1-16x+11+12x-9=0. 257. The theorem of Sturm gives a simple means of establishing the conditions of the reality of the roots. As the real roots are comprised between two limits, L' and +L, the one negative and the other positive, which may be chosen as large as we please, the question reduces to seeking the conditions necessary, in order that from x=-L' to x= =+L the series V, V1, V2, &c., should lose a number of variations equal to the degree of the equation. Supposing this degree to be m, it must then lose m variations. But in order that it may have m variations, it is necessary that it should have at least m+1 terms; and as it can not have more, we are sure that the quantities V, V1, V2, &c., exist to the number m+1, and that they are respectively of the degree m, m—1, m—2, &c. The last, which does not contain x, will then be represented by Vm. When in the polynomial functions of r we substitute very large numbers, positive or negative, for x, we know that the results are of the same sign as if each polynomial were reduced to its first term; therefore, in the present investigation, we need occupy ourselves only with the first term. Let us take the equation V=0 under the ordinary form The first term of V is am; that of the derived polynomial, V1, will be mam-1. With regard to those of the polynomials V2, V3, &c., they are functions composed of the coefficients p, q, &c., determined by the successive divisions in accordance with the rule. Let us represent these functions by G2, G3 ... Gm and write in order the m+1 quantities, The question will be reduced to finding the conditions which will cause the loss of m variations from this series when we pass from x=-L' to x=+L. In order that this may be the case, it must have m variations upon the substitution of L', and m permanences upon the substitution of +L. But in this series the powers of r go on decreasing by unity; consequently, if it has nothing but permanences when x=+L, it will have nothing but variations when x=-L'. Thus, the conditions are reduced simply to such as require this series to have only positive coefficients, that is to say, to the following, G>0, G3>0.... Gm>0. These conditions will never be greater in number than m-1, but they may be less in number, inasmuch as some of the above inequalities may involve the others. EXAMPLE. 258. Find the conditions necessary for the reality of the roots of the equation +qr+r=0. Here we have m=3, and the conditions are only two in number, G>0 and G3>0. 3 To find G, and G3, we calculate V, and V, by successive divisions, as follows: Consequently, the inequalities G,>0, G3>0, become. -2q0, -4q3-27r2>0; observing, however, that the first inequality is embraced in the second, since r is always positive; and changing the signs of the second, we have for the sole condition of the roots of an equation of the third degree, being real, 4q3+277 <0. We have now given so much of the general properties of equations of all degrees, and such modes of proceeding, as will insure their numerical solution in a manner the most certain and infallible, and ordinarily the best. There are, however, many transformations of equations, which, by reducing their degree, or by giving them a particular form, serve to facilitate their solution in certain cases. There are also many general principles applicable to the resolution of equations of the higher orders by the methods in use previous to the discovery of Sturm, which, with these methods themselves, it is desirable to know for many purposes in the application of algebraic analysis to the higher branches of both pure and mixed mathematics, for ulterior improvements in the general theory of equations itself, and even for use in the solution of equations, in some cases, to which they are more conveniently adapted than the method of Sturm. A treatise on algebra could scarcely be regarded as complete without some notice of these. We shall therefore give as extensive an exhibition of them as can in any way be useful in an elementary work like the present, commencing with the well known RULE OF DES CARTES. 259. An equation can not have a greater number of positive roots than there are variations of sign in the successive terms from + to —, or from to +, nor can it have a greater number of negative roots than there are permanences, or successive repetitions of the same sign in the successive terms. Let an equation have the following signs in the successive terms, viz. : +-+---+++−, or +---+-+++· Now, if we introduce another positive root, we must multiply the equation by and the signs in the partial and final products will be x-a, where the ambiguous sign indicates that the sign may be + or according to the relative magnitudes of the terms with contrary signs in the partial products, and where it will be observed the permanences in the proposed equation are changed into signs of ambiguity; hence the permanences, take the ambiguous sign as you will, are not increased in the final product by the introduction of the positive root +a; but the number of signs is increased by one, and, therefore, the number of variations must be increased by one. Hence it is obvious that the introduction of every positive root also introduces one additional variation of sign, and, therefore, the whole number of positive roots can not exceed the number of variations of signs in the successive terms of the proposed equation. Again, by changing the signs of the alternate terms, the roots will be changed from positive to negative, and vice versa (see Prop. VII.). Moreover, by this change the permanences in the proposed equation will be replaced by variations in the changed equation, and the variations in the former by permanences in the latter; and since the changed equation can not have a greater number of positive roots than there are variations of sign, the proposed equation can not have a greater number of negative roots than there are permanences of sign. Let v be the number of variations, v' the number of variations of the transformed equation obtained by changing x into —x. The number of real roots of the equation can not surpass v+v'. Then, if this sum is less than the degree m, the equation will have imaginary roots. The sum + is never greater than the degree, and when it is less the difference is an even number. (See Art. 248.) EXAMPLES. (1) The equation x+3x-412-87x3+400x2+444x-720-0 has six real How many are positive? roots. (2) The equation xa—3.x3 — 15x2+49x−12=0 has four real roots. How many of these are negative? 260. We give next the repetition of a principle already presented, but which may be derived as a direct consequence of the theorem of Sturm. THEOREM OF ROLLE. Let F(x)=0 be an equation which has no equal roots, F'(x) its derived polynomial. We have seen that as r increases, the series of Sturm loses a variation every time that x passes over a root of the equation F(x)=0, and that it can not lose one in any other way. Moreover, we have seen that this variation is lost at the commencement of the series of functions, in consequence of F(x) changing sign, while F'(x) does not; so that F(r) is always of a sign contrary to that of F'(x) for a value of x a little less than the root, and always of the same sign for a value a little greater. Thus, when we ascend from a root r to a root r', which is immediately above r, F(x) must be of the same sign as F'(r) for a value of x a little greater than r, and of a sign contrary to F(r) for a value of x a little less than r'. But in the interval F(r) does not change sign; then F'(x) must change sign at least once; therefore the equation F'(x)=0 has at least one root between r and r'. Let a, b, c, d... g be the real roots of F(x)=0, arranged in order of magnitude, beginning with the largest; and let a,, b1, C1... g1 be the real roots of F(x)=0, disposed in the same manner. We have just seen that these last are comprised, some between a and b, some between b and c, &c.; but as the |