Add the co-efficient of x1 to each of these quotients, and write the sums underneath the quotients which correspond to them. Then divide these sums by each of the divisors, and write the quotients underneath the corresponding sums, taking care to reject the fractional quotients and the divisors which produce them; and so on. When there are terms wanting in the proposed equation, their co-efficients, which are to be regarded as equal to 0, must be taken into consideration. EXAMPLES. 1. What are the entire roots of the equation, A superior limit of the positive roots of this equation (Art. 284), is 13+1 = 14. The co-efficient 48 need not be considered, since the last two terms can be put under the form 16 (x − 3); hence, when x >3, this part is essentially positive. A superior limit of the negative roots (Art. 286), is Therefore, the divisors of the last term which may be roots, are 1, 2, 3, 4, 6, 8, 12; moreover, neither +1, nor will satisfy the equation, because the co-efficient 48 is itself greater than the sum of all the others: we should therefore try only the positive divisors from 2 to 12, and the negative divisors from -2 to 6 inclusively. By observing the rule given above, we have The first line contains the divisors, the second contains the quotients arising from the division of the last term — 48, by each of the divisors. The third line contains these quotients, each augmented by the co-efficient + 16; and the fourth, the quotients of these sums by each of the divisors; this second condition excludes the divisors + 8, +6, and - 3. The fifth contains the preceding line of quotients, each aug mented by the co-efficient 13, and the sixth contains the quo tients of these sums by each of the divisors; the third condition excludes the divisors 3, 2, -2, and -6. Finally, the seventh is the third line of quotients, each aug mented by the co-efficient - 1, and the eighth contains the quo tients of these sums by each of the divisors. The divisors + 4 and 4 are the only ones which give 1; hence, +4 and 4 are the only entire roots of the equation. by the product (x − 4) (x + 4), or x2-16, the quotient will be x2 x+3, which placed equal to zero, gives 4. What are the entire roots of the equation 9x630x5+ 22x2 + 10x3 + 17x2 ·20x40 ? Sturm's Theorem. 298. The object of this theorem is to explain a method of determining the number and places of the real roots of equations involving but one unknown quantity. Let X=0 (1), represent an equation containing the single unknown quantity x; X being a polynomial of the mth degree with respect to x, the co-efficients of which are all real. If this equation should have equal roots, they may be found and divided out as in Art. 269, and the reasoning be applied to the equation which would result, We will therefore suppose X 0 to have no equal roots. = 299. Let us denote the first derived polynomial of X by X1, and then apply to X and X, a process similar to that for finding their greatest common divisor, differing only in this respect, that instead of using the successive remainders as at first obtained, we change their signs, and take care also, in preparing for the division, neither to introduce nor reject any factor except a positive one. If we denote the several remainders, in order, after their signs have been changed, by X2, X3... X, which are read X second, X third, &c., and denote the corresponding quotients by Q1, Q. • Q-1, we may then form the equations Since by hypothesis, X=0 has no equal roots, no common divisor can exist between X and X1 (Art. 267). The last re mainder, will therefore be different from zero, and inde pendent of x. 300. Now, let us suppose that a number p has been substi tuted for x in each of the expressions X, X1, X, ... X~1; and that the signs of the results, together with the sign of X, are arranged in a line one after the other: also that another number 9, greater than p, has been substituted for x, and the signs of the results arranged in like manner. Then will the number of variations in the signs of the first arrangement, diminished by the number of variations in those of the second, denote the exact number of real roots comprised between Р and q. 301. The demonstration of this truth mainly depends upon the three following properties of the expressions X, X1 . . X„, &c. I. If any number be substituted for x in these expressions, it is impossible that any two consecutive ones can become zero at the same time. For, let X1, X2, Xn+1, be any three consecutive expressions. Then among equations (3), we shall find from which it appears that, if X-1 and X, should both become 0 for a value of x, X+ would be 0 for the same value; and since the equation which follows (4) must be = we shall have X+20 for the same value, and so on until we should find X, 0, which cannot be; hence, X1 and X cannot both become 0 for the same value of x. = II. By an examination of equation (4), we see that if X, becomes 0 for a value of x, X-1 and X+1 must have contrary signs; that is, If any one of the expressions is reduced to 0 by the substi tution of a value for x, the preceding and following ones will have contrary signs for the same value. III. Let us substitute a + u for x in the expressions X and X, and designate by U and U1 what they respectively become under this supposition. Then (Art. 264), we have in which A, A', A", &c., are the results obtained by the sub stitution of for x, in X and its derived polynomials; and A1, A1, &c., are similar results derived from X1. If, now, a be a root of the proposed equation X 0, then A = 0, and since A' and à ̧ are each derived from X1, by the substitution of a for x, we have A' A1, and equations (5) become = u2 = U = A'u + A" + &c. U1 = A + A'u + &c. (6). Now, the arbitrary quantity u may be taken so small that the signs of the values of U and U, will depend upon the signs of their first terms (Art. 276); that is, they will be alike when u is positive, or when a + u is substituted for x, and unlike when u is negative or when a u is substituted for x. Hence, If a number insensibly less than one of the real roots of X=0 be substituted for x in X and X1, the results will have contrary signs; and if a number insensibly greater than this root be substituted, the results will have the same sign. 302. Now, let any number as k, algebraically less, that is, nearer equal to ∞, than any of the real roots of the several equations X=0, X1 = 0... X, 0, = be substituted for x in the expressions X, X, X, &c., and the signs of the several results arranged in order; then, let z be increased by insensible degrees, until it becomes equal to h the least of all the roots of the equations. As there is no |