213 



THEORY OF EQUATIONS. 



THEORY OF EQUATIONS. 



211 



proposition is of complex proof, but it begins to occupy its proper 

 place in elementary works, especially on the continent. 



4. \\'e may now refer to STURM'S THEOREM, to Fourier's theorem 

 (given in the article just cited), to Des Cartes' theorem, a very limited 

 particular case of Fourier's, and to Homer's adaptation of, and addition 

 to, the old method of numerical solution by Vieta (an account of the 

 history of this last problem is given in the ' Companion to the Almanac' 

 for 1839). [INVOLUTION AND EVOLUTION.] We have then, since 

 the beginning of this century, a complete theoretical mode of deter- 

 mining the number of roots, real or imaginary, between any given 

 limits; both exceedingly difficult in the complication of the opera- 

 tions which they require. Also, a mode of easy application, though 

 not theoretically perfect, of determining the limits between which 

 the real roots lie ; and a process for the numerical solution which 

 places that question upon the same footing as the common extrac- 

 tion of square, cube, &c., roots; making those extractions them- 

 selves, except only in the case of the square root, much easier than 

 before. In Cauchy's theorem, now beginning to be generally known, 

 and which waa given in the ' Penny Cyclopaedia,' we have a theoretical 

 mode of determining the imaginary roots. And Homer's method has 

 been extended to their computation. But it seldom happens that 

 the actual determination of imaginary roots is required. 



5. The Newtonian method of approximation is in the following 

 theorem. If a be nearly a root of $z=0, and if <f>a : <f>'a be small, 

 then 



<t>a 

 -^a 



is more nearly a root. See APPROXIMATION for the use of this, and 

 TAYLOR'S TIIKOKKM, for a more extensive result. But the use of 

 Homer's method is very much more easy than that of Newton : the 

 former, in fact, includes and systematises the latter. But this remark 

 applies only to algebraical equations : for all others Newton's form just 

 given remains practically unamended. 



6. We refer to the article ROOT for the solution of x + 1 = 0. The 



ing equation, x 1 * + 2 cos 9. if + 1 = 0, admits of complete 

 solution on the same principles. 



7. If Qa and <t>b have different signs, one or some other odd number 

 of roots of <t>x lies between a and 4 : but if they have the same signs, 



no one or an even number of roots lies between a hd 6. Every 

 equation of an odd degree has at least one real root, negative or posi- 

 tive, according as the first and last terms have like or unlike signs. 

 Every equation of an even degree having the first and last terms of 

 unlike signs has at least two real roots, one positive and one negative. 



8. If all the coefficients of <f>x be real, and one of the two, a + 6 V 1, 

 be a root, so is the other : and if all the coefficients be rational, and 

 one of the two, a+ /!>, a and 6 being rational, be a root, so is the 

 other. If there be a rational fractional root, its denominator must be 

 a divisor of the first coefficient, and its numerator of the last, as soon 

 as the equation <f>x = is cleared of fractions. N.B. Among the divisors 

 of a number we reckon 1 and itself. 



9. In the equation a a x* + a, **-i + a, ar- + . . . + o._i x + a. = 0, 

 the sum of all the roota is a, : a^ the sum of the products of every 

 two is^a, : a u , that of the products of every three is o, : a,, and so 

 on. Finally the product of all the roots is + a ,: a^ according as is 

 even or odd. And if r,, r,, . . . r. be the roots, then aj* + ... is the 

 same as o (x - r,) (x - r t ) .... (xr.). 



10. If the preceding expression be called .r,and na^i*-* + ( !) 

 a, a:*-' + . . . , its derived function, be called j/x, we have 



I 



+ ....+ 



and if fr be any rational or integral algebraical function of x, the sum 

 "f""i + +>"i_ +....+ <fr. is the coefficient of the highest power of x in 

 the remainder of the division of # x tyj by $x. 



11. If s. in all cases stand for the sum of the nth powers of the roots 

 of the equation, we have 



and go on up to 



o s. +a 1 s._i+a ll 8 

 after which, in all cases, 



s - r 



0, 



Hence also the coefficients of the expression may be found in terms of 

 s, a, ... s, as soon'as a is given. 



12. All rational symmetrical functions of the roots may be easily 

 expressed in terms 8, S,, &c., and thence in terms of the coefficient* 

 of the expression. 



13. If it be required to find a function tyy the roots of which shall 

 be given function* of those of <t>x, so that in all cases y = Fx, proceed 

 as in finding the highest common divisor of Qx and fxy, and take 



/ the final remainder. But if this final remainder should be of a 



limension than, from the known number of its roots, it ought 



t'. I.. -, it will be a sign that some of the factors introduced in the pro- 



:ess have affected the remainder, and these must be examined and 



removed. The treatment of this case belongs to the general question 

 of elimination, but the following particular cases are almost all that 

 are necessary. 



14. To decrease all the roots of <px by a given quantity, or to make 

 y = xa, or x=y+a, observe that the resulting equation must be 



. y + 





 y> 



where the coefficients <fa, <j>'a, J </>"a, &c., may be the most readily 

 found by the process described in INVOLUTION. The same process 

 may be applied, by using a instead of a, to increase all the roots 

 of <px by a given quantity. It is by this process that the second 

 term of an equation is taken away; thus, the equation being 

 + a L x*~ l + . . . 0, assume 



the sum of the roots of the equation in x being a l : , that of the 

 equation in y will be 0. 



15. To multiply all the roots of an equation by m, multiply its suc- 

 cessive terms, beginning from the highest, by 1 , m, in 2 , m 3 , &c. And 

 to divide all the roots of an equation by m, multiply all the terms by 

 the same, beginning from the lowest. N.B. Terms apparently missing 

 in an equation must never be neglected. Thus a;' 2x t + Sx -1 = 

 ought to be written 



03? + Ox 3 + 3x - 1 = 0. 



This caution is of the utmost importance : in fact no process ought 

 to be applied to any equation without a moment's thought as to 

 whether all the terms be formally written down, and if not, whether 

 the process about to be applied will not require it. 



16. To change the signs of all the roots of an equation, change the 

 signs of the coefficients of all the odd powers, or of all the even powers, 

 as most convenient. 



17. To change an equation into another whose roots shall be reci- 

 procals of the former roots, for every power of x write its complement 

 to the highest dimension. Thus in an equation of the seventh degree, 

 for x write #', for x write z, for x* write X s , and so on : lastly, for x 1 

 write x. N.B. Consider the independent term of the equation as 

 affected by x. From the reciprocal equation can be found the sums 

 of the negative powers of the roots of the original. 



18. The old methods of finding limits to the magnitude of the posi- 

 tive and negative roots of an equation are so rapid that they can lianlly 

 be said to be superseded by those of Sturm or Fourier. In enunciating 

 them we speak of coefficients absolutely, without their signs, when 

 mentioning any increase or decrease they are to receive. 



If A be the greatest of all the quotients made by dividing tho co- 

 efficients by the first co-efficient, no root, positive or negative, is nume- 

 rically so great as A + 1 . And if B be the greatest of all the quotients 

 made by dividing the co-efficients by the last co-efficient, no root, 

 positive or negative, is numerically so small as 1 : (n + 1). Better 

 thus : if L be the first co-efficient, M the greatest, and N the last, signs 

 not considered, then all the roots, numerically speaking, lie between 



M + L 



aud- 



10. If L be the first co-efficient, and M the greatest co-efficient which 

 has a different sign from that of L, no positive root is so great as 

 (M + L) : L. And if L be the last co-efficient and M the greatest which 

 has a different sign, no positive root is so small as L : (M + L). And to 

 apply this to the negative roots, change the signs of all the roots of 

 the original ( 16), and find limits to the positive roots of the new one. 



20. If L be the first co-efficient, M the greatest which has a different 

 sign, and if the first which has .a different sign be in the mth place 

 from the first term exclusive, or belong to the (m + l)th term; then 

 no positive root is so great as 



If 1 be tie number of terms which elapse at the beginning before a 

 change of sign, L the least of their co-efficients, and M the greatest 

 co-efficient of a different sign, any value of x which, being > 1, makes 



positive, is greater than the greatest root : for instance, 



ijPir- 



The following method is very convenient when tho number of terms 

 is large. Divide the whole expression into successive positive and 

 negative lots, A, - B, + C, - D. + . . . p, 1, r, , &c., representing the 

 last exponent of x in each lot. Divide A, B, by afl , and ascertain a 

 value of x, say A, which makes A, B, positive and equal to I. Do 

 the same with lx* + c r D. , which, for x = p. (perhaps no greater 

 than A) becomea m. Repeat the process with mx* + EI - F , and BO 



