5. The Newtonian method of approximation is in the following theorem. If a be nearly a root of 4,x= 0, and if Oa : itia be small, then 4a a - 7 a is more nearly a root. See AirnoxruATIeN for the use of this, and TAYLOR'S THEOREM, 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 Roar for the solution of x -F 1= 0. The following equation, e. + 2 cos + I = 0, admits of complete solution on the same prinUples.
7. If Oa and (jib have different signs, one or some other odd number of roots of ox lies between a and b: but if they have the same signs, either no one or an even number of roots lies between a Slid b. 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 .px be real, and one of the two, a +5 V-1, be a root, so is the other : and if all the coefficients be rational, and one of the two, al- ../b, a and 1. being rational, be a root, so is the other. If there bei rational fractional root, its denominator must be a divisor of the first coefficient, and its numerator of the last, as soon as the equation ctix= 0 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. = 0, the sum of all the roots is - a, : the sum of the products of every two is a, : that of the products of every three is - a, : a and so on. Finally tho product of all the roots 1.3 ± a .: according as n is even or odd. And if r r... r. be the roots, then + ... Is the same as a (x - r,) (x - r,).... (x-r.).
10. If the preceding expression be called 4ix, and (a - I) ..., its derived function, be called fix, have Ix 1 1 1 epx x r, x -r, x- ' and if tfix be any rational or integral algebraiml function of x, the sum + tkr, + . . + is the coefficient of the highest power of x in the remainder of the division of x 4x by ear.
11. if s. in all cases stand for the sum of the nth powers of the roots of the equation, we have 13---sn, + 0, + +2a,= 0, a,s, + a,s, + 3a,= 0, and so on up to + + na. =0,
after which, in all cases, + . +a. sk = 0. Hence also the coefficients of the expression may be found in terms of s, s, s., as soon:as a, is given.
12. All rational symmetrical functions of the roots may be easily expressed in terms 8, 8,, dtc., and thence in terms of the coefficients of the expression.
13. If it be required to find a function 4y the roots of which shall be given functions of those of epx, so that in all cases y=rx, proceed as in finding the highest common divisor of q,x and Fx-y, and take for the final remainder. But if this final remainder should be of a higher dimension than, from the known number of its roots, it ought to be, it will be a sign that some of the factors introduced in the pro cess 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= or x=y +a, observe that the resulting equation must be 4190a where the coefficients (pa, lict, A 4"a, fie., may be the most readily found by the process described in INvoLuTrosi. The same process may be applied, by using - a instead of a, to increase all the roots of epx by a given quantity. It is by this process that the second term of an equation is taken away; thus, the equation being + + 0, assume 1 a, the sum of the roots of the equation in x being -a, : a, 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,&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 3x-I=0 ought to be written + + - 2.ca + 0s.. + + 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.