1 Proceed with V and V, as in finding their greatest common divisor, being careful to change each sign in the successive remainders before they are used as divisors, and to continue each division until the remainder is of less dimensions in x than the corresponding divisor; then if Q1, Q2, Q3, etc., stand for the successive quotients, and -V2, - V39 V1, etc., for the corresponding remainders, the equations V=V1Q1—V2, V1 = V2Q2 —V3, V2 = V3Q3 − V4, etc., (3), will be obtained. 1 Calling the remainders in (3), after their signs are changed, the modified remainders, write V and V1, together with the modified remainders, in their proper order, and the series V, V1, V2, V3 . ... N, (4), will be obtained; noticing that N is used for the last modified remainder. 1 If the equations V = 0 and V1 = 0 can be satisfied by the same value of x, or if V = 0 has equal roots, it is clear that N must be a function of x, which is the greatest common divisor of V and V1; consequently, those values of a which satisfy the equation N = 0 will evidently reduce each term of (4) to 0. Also, those values of x which reduce any two or more adjacent terms in (4) to 0, will reduce all the terms to 0, and will result from the equal roots of V = 0. Thus, let the equations V1 = 0 and V, = 0 be satisfied by the same value of ; then it is clear, from (3), that the same value of reduces each term of (4) to 0. 2 Hence, there is no value of x that is not one of the equal roots of V = 0, which can reduce two successive terms of (4) to 0. x If V = 0 has no equal roots, it follows, from what has been done, that N is independent of x, and that there is no value of which can reduce two successive terms of (4) to 0. If a is such a value of x as reduces one of the terms of (4), between V and N, to 0; then, if a is not one of the equal roots of V = 0, the terms which are adjacent to the term reduced to 0 in (4) shall have contrary signs. 3 = Thus, suppose that a, when put for a, gives the equation V1 = 0; then the equation V, V,Q, — V, of (3) is reduced to V, V; consequently, V, and V, in (4) have unlike signs. 5 2. Let p and q be two numbers, such that p is greater than q, or that p = q +r, r being positive; then substitute q and p successively for x in the terms of (4), and represent the number of their variations of signs which correspond to q and p by m and m'; and the number of units in m - m' will exactly equal the number of unequal real roots of (1) or V = 0, which lie between the numbers q and p. If infin. and ∞ = + infin. are put for q and p, it is plain that the terms in (4) will be reduced to their first terms nearly, or to those which contain the highest powers of x in them; consequently, the number of units in mm' will exactly equal the number of unequal real roots of V=0. Remark. Any factor that is suppressed or introduced (as in finding the greatest common divisor), to give integral quotients in obtaining (3), must be positive, so that the signs of the terms of the remainder may not be changed. PROOF OF THE THEOREM. To simplify the reasoning, we shall conceive a to increase by very small increments from q to p, and to pass through the roots of V = 0, which may lie between q and p. 1. If x, during its increase, passes through one of the unequal roots of V = 0, then a variation, which will exist between the signs of V and V1, just before a passes through the root, will be changed into a permanence just after the passage; consequently, a variation between the signs of V and V1 in (4) will be lost. 1 2. If a passes through n' equal roots during its increase; then, if a denotes one of these roots, it is clear that (a) must be a factor of V; consequently, if V' stands for the remaining factor, we shall have V = (x − a)"' × V'. If we puty for x in (a)" × V', and represent the coeffi cient of y resulting from the substitution of x + y for x in V' by R; then we shall get n'(x − a)"'—1V' + (x − a)"'R for the coefficient of y in the result, which will be the first derived function of V. Hence, V and V, will be represented by (x-a)"'V' and n'(x − a)"'-1V'+(x−a)"'R; consequently, by division, we shall get V V1 = (x-a)"'V' -1 n'(x − a)”' −1V' + (x − a)"'R n'(x-a)' = (x-a)V' n'V' + (x − a)R small. x a = n' very nearly, when xa is very Because a is negative, when x is a very little less than a, and positive when ≈ is a very little greater than a, it follows, that when a increases from a value, a very little less to V ར་ one, a very little greater than a, the sign of will be changed from - to; consequently, a variation between the signs of V and V, in (4) will be lost by the passage of x through the n' equal roots of V = 0. 1 — — 1 Because V = (x − a) (x − a)n'-1V' and V1 = [n'V' + (x − a)R] × (x − a)"'-1, it follows that they have (-a)"'-1 for a factor, which is also a factor of all the remaining terms of (4); and it is evident that the loss of a variation between the signs of V and V1 is due to the change of the sign of x a (and not to that of (x — a)"'-1) in V, resulting from the passage of a through the n' equal roots of V = 0; indeed, because the change of the sign of (a)"'-1 changes the (x signs of all the terms of (4), it clearly can not affect the number of variations of their signs. Since all those values of a which reduce N, or any two or more successive terms of (4) to 0, have been shown to result from equal roots of V = 0; it plainly follows, from what has been done, that such values of x can not affect the number of variations in (4). 4 3. If a passes through a value which is not one of the equal roots of V = 0, and reduces one of the terms between V and N, as V, in (4) to 0, then, according to what has been shown, the terms V, and V, adjacent to V, will have contrary signs; consequently, since the passage of x through the root of V=0 can not affect the signs of V, and V, it follows that there will be one variation of signs between V, and V, when V, 0, and that one variation of signs between the signs of V, V4, and V, will exist just before and after V, is reduced to 0, and of course the passage of 7 through the root of V1 = 0 can not affect the number of variations of signs among the terms of (4). 4 = Hence, if a passes through a value which is not one of the equal roots of V = 0, and reduces two or more of the terms in (4) to 0; then, since the terms reduced to 0 can not be adjacent to each other, and that the signs of the terms adjacent to the terms thus reduced are unlike and not changed by the reduction of the terms to 0, it follows, from what has been shown, that the passage of a through any such value can not affect the number of variations among the signs of the terms in (4). 4. Neglecting the multiplicity of a root, or considering such roots as single roots; then, since a variation is lost among the signs of (4), when a during its supposed increase from q top reduces V to 0, or passes through a real root of V = 0, and that those values of a which reduce one or more of the remaining terms of (4) to 0 can not affect the number of variations of signs in (4), it follows that the number of variations of signs lost in (4) is exactly equal to the number of unequal real roots of V = 0 which lie between q and p; consequently, the theorem is true. To perceive the use of the theorem, take the following EXAMPLES. 1. To find the number of real roots of a3 3x 1 = 0. Here we have 32-3 for the first derived function, and omitting the useless factor 3 we may take 2-1 for it; and dividing a 3x-1 by x2 - 1, we have the remainder, whose signs being changed, give 2x + 1 for the first modified remainder; also multiplying — 1 by 4, and dividing the product by 2x + 1, we have the remainder, whose sign being changed, gives 3 for the second modified remainder; hence, collecting the results, we get a 3x1, x2-1, 2x + 1, 3, for the representatives of the terms in (4). Putting 2 for x in the preceding functions, their signs are,,,+, three variations, also putting 2 for a in the same functions, the signs are +, +, +, +, no variation; consequently, since m = 3 and m' 0, we have mm' = 3 the number of real roots of the equation which lie between 2 and +2. Hence, because the equation can have only three roots, it follows that the roots are all real and have 2 and 2 for inferior and superior limits. = Again, putting -1 for a in the functions, they become 1, 0, — 1, and 3, whose signs, by omitting 0, become +, −, +, two variations; consequently, since 2 gives three x = variations, it follows that one of the roots lies between 2 and - 1, and of course -1 must be the first figure of one of the roots. Also, because = 0 gives one variation, it follows that one of the roots lies between 1 and 0; and since x = − 0.4 gives two variations and 0.3 gives one variation, it follows that 0.3 are the first figures of this root. Similarly, because 1 gives one variation, and that = 2 gives no variation, it follows that 1 is the first figure of the remaining root; thus, -1, 0.3, and 1 are the principal parts of the three roots. pared to (6) of Sec. XV., gives n = 1, y= 3, a = 0, Hence, from (since y' is here=y) (c) of the same Sec., we 0.347291 nearly, whose first five decimal places are correct. Again, if we multiply the terms of the given equation by -1, the product is easily reduced to the form æ2 — x-1 = 3; consequently, from (c) we (as before) get = ± √3 + + + etc. = ± 1.73205 + 0.1666. 6 8×913 162 1 F 3 1 0.02405 |