from the first to the second; one permanence from the second to the third; and one variation from the third to the fourth: thus making again two variations and one permanence, The number of variations of signs therefore, in this case as well as in the former, is the same as that of the positive roots; and the number of permanencies, the same as that of the negative roots. Corol. Whence it follows, that if it be known by any means whatever, that an equation contains only real roots, it is also known how many of them are positive, and how many negative. Suppose, for example, it be known that, in the equation x+3x1 23x3-27x2 + 136x — 120 = 0, all the roots are real: it may immediately be concluded that there are three positive and two negative roots. In fact this equa. tion has the three positive roots x = 1, x = 2, x = 3; and two negative roots, x = 5. - 4, x = If the equation were incomplete, the absent terms must be supplied by adopting cyphers for coefficients, and those terms must be marked with the ambiguous sign. Thus, if the equation were -- all the roots being real, and the second term wanting, it must be written thus: x50x1 20x330x2 + 19x 30= 0. Then it will be seen that, whether the second term be positive or negative, there will be 3 variations and 2 permanencies of signs and consequently the equation has 3 positive and 2 negative roots. The roots in fact are, 1, 2, 3, 1, — 5. This rule only obtains with regard to equations whose roots are real. If, for example, it were inferred that, because the equation x2+2x+50 had two permanencies of signs, it had two negative roots, the conclusion would be erroneous: for both the roots of this equation are imaginary. THEOREM III. Every equation may be transformed into another whose roots shall be greater or less by a given quantity. In any equation whatever, of which x is unknown, (the equations A, B, C, for example) make x = z + m, z being a new unknown quantity, m any given quantity, positive or negative: then substituting, instead of x and its powers, their values resulting from the hypothesis that x =z+m; so shall there arise an equation, whose roots shall be greater or less than the roots of the primitive equation, by the assumed quantity m. VOL. II. 15 Corol. The principal use of this transformation is, to take away any term out of an equation. Thus, to transform an equation into one which shall want the second term, let m be a so assumed that nm a = 0, or m = n being the index of the highest power of the unknown quantity, and a the coefficient of the second term of the equation, with its sign changed then if the roots of the transformed equation can be found, the roots of the original equation may also be found, because x = x+ a THEOREM IV. Every equation may be transformed into another, whose roots shall be equal to the roots of the first multiplied or divided by a given quantity. 1. Let the equation be 23+ az2 + bz + c = 0: if we put fz = x, or z = the transformed equation will be 3+ fax +fbx +ƒ3c 0, of which the roots are the respective products of the roots of the primitive equation multiplied into the quantity f. By means of this transformation, an equation with frac. tional quantities, may be changed into another which shall be free from them. Suppose the equation were 23+ bz d h k ghk' g + the trans + =0: multiplying the whole by the product of the denominators, there would arise ghkz3 + hkaz2 + gkbż + ghd = 0: then assuming ghkz =x, or z formed equa. would be x3+hkax2 + g3k3hbx + g3k3h3d = 0. The same transformation may be adopted, to exterminate the radical quantities which affect certain terms of an equation. Thus, let there be given the equation z3 + az2 √ k + bz + c✔k: make z k = x; then will the transformed equation be a3 + akx2 + bkx + ck2 = 0, in which there are no radical quantities. 2. Take, for one more example, the equation z3 + az3 + bx + c = 0. Make; then will the equation be are equal to the quotients of those of the primitive equations divided by f. It is obvious that, by analogous methods, an equation may be transformed into another, the roots of which shall be to those of the proposed equation, in any required ratio. But the subject need not be enlarged on here. The preceding succinct view will suffice for the usual purposes, so far as relates to the nature and chief properties of equations. We shall therefore conclude this chapter with a summary of the most useful rules for the solution of equations of different degrees, besides those already given in the first volume. I. Rules for the Solution of Quadratics by Tables of Sines and Tangents. 1. If the equation be of the form x2+px=q: 2. For quadratics of the form x2-px=q. 2 Make, as before, tan a == ✔g: then will 3. For quadratics of the form x2+px=—q• 4. For quadratics of the form x2-px=—q. In the last two cases, if✔q exceed unity, sin á is ima ginary, and consequently the values of x. 1 The logarithmic application of these formulæ is very simple. Thus, in case 1st. Find a by making A 10+log 2+ log q-log p-log tan A. Then log x = -(log cot A+ log q-10). Note. This method of solving quadratics, is chiefly of use when the quantities p and q are large integers, or complex fractions. II. Rules for the Solution of Cubic Equations by Tables of Sines, Tangents, and Secants. 1. For cubics of the form x3+px±q=0. Then x cot 2a. 2/p. 2. For cubics of the form a3—px±q=0. Make sin B = 2✓p..... tan Atan B. Then cosec 2a. 2/p. Here, if the value of sin B should exceed unity, в would be imaginary, and the equation would fall in what is called the irreducible case of cubics. In that case we must make cosec 3a = P 2p and then the three roots would be : sin (60+ a) · 2√3p. If the value af sin в were 1, we should have B=90°, tan A = 1; therefore a == 45°, and x = 2p. But this The second solution would give would not be the only root. 90° cosec 3A 1: therefore a = ; and then 3 r = ± sin 30° . 2✓1⁄2p = ± √ P Here it is obvious that the first two roots are equal, that their sum is equal to the third with a contrary sign, and that this third is the one which is produced from the first solution*. The tables of sines, tangents, &c. besides their use in trigonometry, and in the solution of the equations, are also very useful in finding the value of algebraic expressions where extraction of roots would be otherwise required. Thus if a and b be any two quantities, of which a is the b b greater. Find z, z, &c. so, that tan xv, sin z = V α logv(a2-b2)-log a log sin y-log blog tan y. = logy(a+b)-log a+log sec u log blog cosec u. a The first three of these formula will often be usefel, when two sides of a right-angled triangle are given, to find the third. In these solutions, the double signs in the value of x, relate to the double signs in the value of q. N. B. Cardan's Rule for the solution of Cubics is given in the first volume of this course. III. Solution of Biquadratic Equations. = Let the proposed biquadratic be x + 2px3 qx2 + rx +s. Now (x2+px + n)2= x2+2px3+ (p2+2n) x2 + 2pnx +n2: if therefore (p2 + 2n) x2 + 2pnx + n2 be added to both sides of the proposed biquadratic, the first will become a complete square (x2 + px + n), and the latter part (p2 + 2n + q) x2 + (2pn + r) x + n2+s, is a complete square if 4 (p2 + 2n +q) · (n2 + s) = 2pn+r2; that is, multiplying and arranging the terms according to the dimensions, of n, if 8n3 +4gn2 + (8s4rp) n + 4qs + 4p's — 20. From this equation let a value of n be obtained, and substituted in the equation (x2 + px + n)2 = (p2 + 2n + q)x2 + (2pn + r) x + n3 + s ; then, extracting the square root on both sides is positive; x2+px+n=± {√(p2+2n+q)x+√ (n2+s) { when2pn+r or x2+px+n=± }√(p2+2n+q)x−√(n2+s) { is negative. And from these two quadratics, the four roots of the given biquadratic may be determined*. Note. Whenever, by taking away the second term of a biquadratic, after the manner described in cor. th. 3, that fourth term also vanishes, the roots may immediately be ob tained by the solution of a quadratic only. A biquadratic may also be solved independently of cubics, in the following cases: 1. When the difference between the coefficient of the third term, and the square of half that of the second term, is equal to the coefficient of the fourth term, divided by half that of the second. Then if p be the coefficient of the second term, the equation will be reduced to a quadratic by dividing it by x2+1px. 2. When the last term is negative, and equal to the square of the coefficient of the fourth term divided by 4 times that of the third term, minus the square of that of the second: then to complete the square, subtract the terms of the proposed biquadratic from (px), and add the remainder to both its sides. * This rule for solving biquadratics, by conceiving each to be the dif ference of two squares, is frequently ascribed to Dr. Waring; but its original inventor was Mr. Thomas Simpson, formerly Professor of Mathematics in the Royal Military Academy. |