8th. What is the present value of an annuity of 350 £. assigned for 8 years at 4 per cent. interest ?..... .Ans. 2356 £. 9 sh. 2 d. 9th. The number of settlers in a colony being 20000; if the yearly rate of increase of the population is part, how many inhabitants will the colony contain in 100 years ?................Ans. 14489276. 10th. A town contains 20000 inhabitants; the population having regularly increased yearly, what was the number of inhabitants 10 years ago ?... .............Ans. 14882. SECTION XI. OF THE COMPOSITION OF EQUATIONS. 226. The only equations of which complete solutions have been obtained are those of the first and second degree. But general properties which lead to the resolution of numerical equations of any degree whatever have been discovered. These properties are connected with a particular form to which every equation can be reduced. An equation of any degree, m, involving one unknown quantity, ought, when complete, to contain all the powers of the unknown quantity from that of the degree m to the first inclusive, each being multiplied by known quantities, and, besides these, a term containing only known quantities. The equation of the fourth degree, for example, ought to contain all the powers of r from the fourth to the first inclusive. If the same power of x is found in several terms, they must be reduced into one. Then all the terms of the equation being transposed to the left member, and the sign of the first term being rendered + by the change, if necessary, of the signs of all the other terms, the equation has the form, nx++px3+qx2+rx+s=0, in which the letters p, q, r, s may represent negative as well as positive numbers. Dividing all the terms by n, in order that the coefficient of the first term = 8 may be 1, and making ?=P, P=P, 1=Q, R, S, the preceding equation becomes n n x++Px3+Qx2+Rx+S=0. In the following pages it is always supposed that equations are reduced to this form. Hence the general equation of any degree, m, may be represented as follows: in which the interval marked by points is filled up when a particular value is given to the exponent m. For the sake of brevity, the first member of the equation xTM+PxTM-1+ &c. 0 is sometimes denoted by a capital X. Every expression, whether real or imaginary, which, being substituted for x in an equation prepared in this manner, renders its first member equal to zero, and by consequence satisfies the equation, is said to be a root of the equation. The meaning of the word root, in this definition, is more general than when it is employed to describe the result of the extraction of some root of a quantity, a. 227. The root of any equation of the form X=0 being represented by a, the first member of this equation is exactly divisible by the binomial x-a. For, since a is a value of x, if a is substituted for x in the general equation X=0, a"+Pa"-1+Qa"-2 1-2 so that the proposed equation is identically the same as x+Px1+QxTM-2 -a-Pa-1-QaTM-2 The quantities x”—a", x′′-1-aTM-1, x”—2—aTM ..... +T(x-a)=0. divisible by x-a (Art. 22), it is evident that the first member of the equation xTM—aTM+Р(xTM-1—a" - !) + . . . . . +T(x−a)=0 is divisible by this quantity; and that the first member of the proposed equation X=0 is therefore divisible by x-a, which was to be proved. To form the quotient it is sufficient to substitute for the quantities r”—a", x-1-a-1,. . . . x-a, the quotients which are obtained from the division of these quantities by x-a, and which are, respectively, Arranging the results according to the powers of x, +P 3 Xh + Pa +Q Denoting a+P by P', a2+Pa+Q by Q', &c. this equation (which is of the degree m-1) can be put under the form "+P'xTM-2+Q'x*¬3+, &c. =0, or X'=0. -2 -3 Whence a+Px”¬1+QxTM-2+, &c. =(x−a)(x2-1+P′xTM-2+Q'xTM¬3+, &c.), and the proposed equation can be verified in two ways, namely, by making x-a=0, or by making a-1+P2x2-2+Q'xTM-3+, &c. =0. -2 -3 If the equation x1+P'x2+Q'x"¬3+, &c. =0 has a root, b, its first member is divisible by x-b, and there is found as before, x3¬1+P′x”¬2+Q'x®-3 +, &c. = (x—b)(xTM−2+P''xTM¬3+Q′′xTM→++, Consequently "+Px"¬1+Qx" 2+, &c. =(x−a)(x—b)(x2+P"x-3+ Q''x"++, &c.) -4 The equation X=0 can therefore be verified by x—a=0, x—b=0, 2”−2+P′′x”−3+Q′′x”¬4+, &c. =0. &c.) If the last of these equations has a root, c, its first member can be again decomposed into two factors, x-c, xTM¬3+P'''xTM¬++Q′′"'xTM−5+, &c. and x"+Px”¬1+Qx"−2+, &c. =(x−a)(x—b)(x−c)(xTM¬3+P'''x”÷÷Q′′′′x®-3, +&c.) -2 Whence the proposed equation can be verified in four manners; namely, by making in turn x—a=0, x—b=0, x—c=0, xTM3¬3+P'''xTM¬÷+Q′′x*−5+, &c. =0. By continuing this process of decomposition, factors of the degrees m-4, m-5, m-6, &c. are obtained in succession; and if each of these factors, when made equal to zero, is susceptible of a root, the first member of the equation X=0 is reduced to the form (x−a)(x—b)(x−c)(x−d) . . . (x−1); that is, it is decomposed into as many factors of the first degree as there are ones in the exponent m of its degree. The equation ”+Px”¬1+Qx”~2 + · ··· +Tx+U=0 is, in this case, capable of verification in m ways, namely, by making x-a=0, x-b=0, x-c=0, x-d=0,.. · · · · · . . . x —1—0. It is necessary to remark, that these equations ought not to be regarded as true, except in turn, and that to suppose them to have place at the same time leads to manifest contradictions. For from x-a=0 is obtained x=a, and from x-b=0 is obtained r=b, consequences which cannot be reconciled when a, b are unequal quantities. In effect x, in each of these equations, represents the unknown quantity, as it may do in any two unrelated equations whose roots, although equal to x, are not equal to each other. 228. The first member of the equation "+Pa+QxTM-2+ ... Tx+U=0, being decomposed into m factors of the first degree, x—a, x—b, x—c, x—d, x-1, it cannot be divided by any other expression of the first degree; for if its division by a binomial x-α, different from all of these, were possible, then, as in the case of division by x-a, x+PxTM-1+QxTM−2+, &c. =(x−a)(xTM-1+pxm−2+qxTM−3+, &c.) and consequently (x-a)(x-b)(x-c)(x-d)... (x-1)=(x−a)(xTM-1+px3-2+9xm¬3+, &c.). Changing x into a in this expression, (a-a) (a-b)(a-c)(a-d)... (a-1)=(a—a) (aTM1+pa"-2+ga-3+, &c.) Now the factor a―a=0, and therefore the second member of this equation vanishes; but this is not the case with the first member, which is the product of factors all different from zero, so long as a differs from each of the roots a, b, c, d, . . . ..7; the supposition, therefore, is not true. Therefore an equation of any degree whatever cannot have more binomial factors of the first degree than there are ones in the exponent of its degree, and, consequently, it cannot have a greater number of roots. In this investigation it is assumed that every algebraic equation has at least one root, real or imaginary. If some of the quantities a, b, c.... are equal to each other, the number of different values of r is less than m. The degree of the equation is, however, always taken to indicate the number of roots, it being at the same time understood that two or more of these roots may be equal. 229. If an equation is considered as the product of factors x-a, x-b, x-c, x-d.... in number equal to the exponent of its degree, it has the form of the product of Article 139, with this difference, that its terms are alternately positive and negative. Taking four factors, x-a, x-b, x-c, x-d, the equation (-a)(x—b) (x-c)(x-d)=0 is expressed by, xax3+ab x2-abc | x+abcd=0. -b +ac +bd Hence, the second terms of the binomials x-a, x-b, x-c, x-d, &c. being the roots of the equation taken each with a contrary sign, the properties proved in the case of the equation of the second degree subsist in the present instance, as follows: The coefficient of the second term, taken with a contrary sign, is equal to the sum of the roots. The coefficient of the third term is equal to the sum of the products of the roots multiplied together two by two. The coefficient of the fourth term, taken with a contrary sign, is equal to the sum of the products of the roots multiplied together three by three; and so on for the other coefficients, care being taken to change the signs of the coefficients of the even rank. The last term (subject to the law which attaches to the sign of the terms of an even rank) is equal to the product of all the roots. By making equal to zero the product of the three factors r-5, x+4, x+3, the equation a3+2x2-23x-60-0 is formed, the roots of which are +5, -4, -3. The sum of these roots is 5+ (−4)+(−3)=-2. The sum of their products, two by two, is (+5)×(-4)+(+5)×(−3)+(-4) × (−3)=—20+(−15)+12=-23, and their product is (+5)x(-4) (-3)=+60. These results, -2, −23, +60, can be deduced from the coefficients 2, -23, -60 by changing the signs of the 2d and 4th terms. If the product of the factors x-2, x−3, x+5, is made equal to zero, the equation obtained is x3-19x+30=0, which wants the second term or term in, and this because the sum of the roots, which taken with contrary signs form the coefficient of the 2d term, is 2+3-5=0; that is, because the sum of the positive is equal to the sum of the negative roots. 230. An equation being considered as formed of the product of many factors of the first degree, it has been proved that the equation cannot have more simple factors than the number marked by the exponent of its degree. But if the simple factors x-a, x-b, &c. are combined two by two, quantities of the second degree are formed, which are also factors of the equation, and m(m-1) of which the number is expressed by the formula so that if the equation is of the fourth degree the number of factors of the second degree or 6. is 4x3 9 2 9 1.2 If the simple factors are combined three by three, the number of factors m(m—1)(m—2) of the third degree is expressed by the formula 1.2.3 x+PxTM-1+QxTM-2+ ... +Tx+U=0, which, for being an equation which has a root a, if the equation is divided by x-a, the quotient is +Pam-2 +Qa-3=0. +T This equation, which is depressed to the degree m-1, must contain the remaining roots of the proposed equation. The coefficients of the depressed are formed from the coefficients of the proposed equation, as follows: The coefficient of the first term in the depressed is equal to the coefficient of the first term in the original equation. The coefficient of the second term is equal to the coefficient of the first term multiplied by the root a, plus the coefficient of the second term of the original equation." The coefficient of the third term is equal to the coefficient of the second term of the depressed equation multiplied by the root a, plus the coefficient of the third term of the original equation. The coefficient of the mth terin of the depressed equation is equal to a times the coefficient of the preceding or (m-1)th term of the depressed equation, plus the coefficient of the mth term of the original equation. One root of the proposed equation must be given when the depressed equation is required. 1st example. Let a root of the equation, 1525—19x++6x3+15xa—19x +6=0, be 6; required the depressed equation, or the equation containing the other roots? The coefficient of the first term being 15, 15x is the first term of the depressed equation. 15×6+(-19)=9-19=-10, .. -10x3 is the second term, 15 × 6+(-19)=9-19=-10, .. -10 is the last term, The formation of the coefficients of the depressed from the coefficients of the proposed may be represented concisely, thus: Coefficients of the proposed equation Corrections 15, 19, +6, +15, −19 Coefficients of the depressed equation 15, -10, 0, +15, -10 2d example. One root of the equation x3-3x2-10x+24=0 being 2, it is required to find the equation in a2 involving the other roots? Ans. x-x-12=0. 3d example. If 3 is a root of the equation x-11x2+14x-24=0, what is the depressed equation containing the other roots? Ans. x+3x-2x+8=0. 4th example. One root of the cubic equation x3-7x2+36=0 is 3, what are the other roots? Ans. 6 and -2. SECTION XII. OF THE TRANSFORMATION OF EQUATIONS. 232. In the transformation of equations the object is to change one equation into another whose roots shall have a given relation to the roots of the first equation. The principal cases of transformation are, 1st. To change the signs of the roots of a given equation. This amounts to the finding of an equation whose roots are those which would be obtained by changing the signs of the roots of the given equation; therefore if x is the unknown quantity of the given equation and y that of the transformed equation, the relation x=-y must subsist between them. Whence it is necessary to replace x by -y in the given equation; for it is evident that if a quantity, a, is a root of one of the equations, the quantity -a must be a root of the other. Let the given equation be x"+Px"¬1+QxTM¬2+, &c. =0 (A). If -y is substituted for r the absolute values of the coefficients of the equation in y are evidently the same as those of the equation in x, but the signs of the coefficients of the odd powers are contrary. If m is even the first term of the transformed equation is y", but if m is odd the first term is -y. By changing the signs of all the terms of the transformed equation (which is permitted) the first term is rendered positive; in this case the coefficients of the even powers of the proposed and transformed equations have contrary signs. For example, if the proposed equation is x-5x2-5x+1=0, the transformed equation, which has the same roots, but of contrary signs, is x3+5x2-5x-1=0. 233. 2d. To form an equation whose roots shall be the reciprocals of the roots of a given equation. x"+Px+QxTM-2....+Tx+U=0 (A). Two quantities whose product is 1 are reciprocals each of the other; hence, if the roots of equation A are a, b, c, . . . the roots of the transformed equation must be 1 1 a' b1 • .... Consequently it is sufficient in A to change x |