III. Substitute this binomial and its powers, for the unknown quantity and its powers in the given equation, and there will arise a new equation wanting its second term '. EXAMPLES.-1. To transform the equation x3 +12 x2—8 x—9 =0, into an equation wanting its second term. 12 OPERATION. First -=+4. Let y-4=x. Then, x=(y-43) y3—12 y2+48 y— 64. Explanation. I first divide the coefficient 12 of the second term by the index 3; the quotient 4 I annex to a new letter y, first changing its sign from + to, making y-4; this quantity and its powers are next substituted for x and its powers, as in the two foregoing rules; then adding the like quantities together, the sum is the equation y3 *—56 y + 151=0, wanting its second term, as was proposed. 2. To destroy the second term from the equation xa—ur3+ bx2-cx+d=o. α First, is the coefficient of the second term divided by the 4 index of the first. a Let y be the new letter, then by the rule, y+2=x, whence ! This rule is necessary to the solution of cubic and biquadratic equations; and the truth of it will appear from an attentive examination of the process in ex. 1. The third, fourth, and fifth, &c. terms may be exterminated from any +do, which, properly contracted, becomes y+b—? 3 a* 16 4 4 3. Given x2-4x+8=0, to exterminate the second term. -4 Thus, === 2 =- ·2; then let y+2=x, and proceed as before. 4. Given x2+10x-100=o, to destroy the second term. 5. To exterminate the second term from x3-3x2+4x—5—0. 6. Let the second term be taken away from the equation x++24x3-12 x2+4x-30=0". 7. To take away the second term from the equation x350 x 40 x3-30 x2+20 x-10=o. 38. To multiply the roots of an equation by any given quantity, that is, to transform it into another, the roots of which will be any proposed multiple of those of the given equation. RULE I. Take some new letter as before, and divide it by the given multiplier. II. Substitute the quotient and its powers, for the unknown quantity and its powers, in the given equation, and an equation equation, but these transformations being less useful and more difficult than the above, we have in the text omitted the rules: in general, to take away the second term requires the solution of a simple equation; to take away the third term, a quadratic; the fourth term, a cubic; and the nth term requires the solution of an equation of n-1 dimensions. See the note below. This contraction consists in the reducing of the fractional coefficients of the same powers of y to a common denominator, and then adding or subtracting, according to the signs; putting the coefficients of the same power of y under the vinculum, &c. &c. In like manner, to take away the third term from the equation x3-ax3 +bx-co, we assume y+e=x, where e must be taken such that (suppos ing n the index of the highest power of x) n. = n-l -n-1, ae+b=o. In which case a quadratic is to be solved; and in general, to take out the mth term, by this method, an equation of m-1 dimensions must be solved, as was observed in a preceding note. See Wood's Algebra, p. 141. will thence arise, whose roots are the proposed multiple of those of the given equation. RULE I. Assume some new letter as before, and place the given quantity under it, for a denominator. II. Substitute this fraction and its powers, for the unknown quantity and its powers respectively, in the given equation, and a new equation will arise, having its roots respectively equal to the given equation multiplied by the given quantity i. EXAMPLES.-1. To transform the equation 2+5x-2=0, into another, the roots of which are 10 times as great as those of the given equation. 2. Let the roots of 3 x3-12 x2 + 15x-21=0, be multiplied by 3. Therefore (+5y-21, or) y3 — 12 y2+45y 9 -189=0, the equation required. i This rule requires neither proof nor explanation; it is sometimes useful for freeing an equation from fractions and radical quantities. * Hence it appears, that to multiply the roots of an equation by any quantity, we have only to multiply its terms respectively by those of a geometrical progression, the first term of which is 1, and the ratio the multiplying quan 4. Let the roots of x2 -3x+4=0, be doubled. 5. Let the roots of x3 + 12 x2—20x+50=o, be multiplied by 100. 39. To transform any given equation into another, the roots of which are any parts of those of the given equation. RULE I. Assume a new letter as before, and let it be multiplied by the number denoting the proposed part. II. Substitute this quantity and its powers, for the unknown quantity and its powers, in the given equation; the result will be an equation, the roots of which are respectively the parts proposed of those of the given equation'. EXAMPLES.-1. Let the roots of x-x-5=0, be divided 2. Let the roots of x3 +7 x2-29x+2=o, be divided by 5. 3. Given xa—2 x2—3 x+4=o, to divide its roots by 8. 40. To transform an equation into another, the roots of which are the reciprocals of those of the given equation. RULE I. Assume a new letter, and make it equal to the reciprocal of the unknown quantity in the given equation. tity, thus, in ex. 1. the roots of the equation are to be multiplied by 10; wherefore multiplying the given equation x2 + 5x- 2=0 by the geometrical progression 1 10 100 The product is x2 + 50 x—200=0, as above, where y in the above example answers to x in this; and the like in other cases. This rule is equally evident with the foregoing; and in like manner, the roots of an equation are divided by any quantity, by dividing its terms by those of a geometrical progression, whose first term is 1, and ratio, the said quantity: Thus, ex. 1. to divide the roots of x2 Divide its terms respectively by 1 5 =0 by 3, x 3 9 course to this rule, to exterminate surds from an equation. II. Substitute the reciprocal of this letter and its powers, for the unknown quantity and its powers, in the given equation; the result will be an equation, having its roots the reciprocals of those of the given equation. EXAMPLES.-1. Let the roots of x-2x2+3x-4=0, be transformed into their reciprocals. 2. Let the roots of x2+10x-25=0, be changed into their reciprocals. 3. Change the roots of x3-ax2+bx-c=o, into their reciprocals. -3 4. Change the roots of x+ar"——bx" —2 + cx" — 3 —d=o, into their reciprocals. 41. To transform an equation into another, the roots of which are the squares of those of the given equation. RULE. Assume a new letter equal to the square of the unknown quantity in the given equation; then by substituting as in the preceding rules an equation will arise, the roots of which are the squares of those of the given equation. EXAMPLES.-1. Let the roots of the equation x2+9x—17=0, be squared. Assume y=x2 Then x2=y +9x=+9√Y -17=..... -17 Whence y+9y-17=0, the equation required TM. The roots of the proposed equation are 1.6 and 10.6: those of the |