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 bo 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 a° write xi, for x write x", for write x", and so on : lastly, for 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 tho 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to the magnitude of the posi tive and negative roots of an equation are so rapid that they can hardly be said to be superseded by those of Sturm or Fourier.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 the co efficients by the first co-efficient, no root, positive or negative, is nume rically so great as +1. And if a be the greatest of all the quotients made by dividing the coefficients by the last co-efficient, no root, positive or negative, is numerically so small as I : 1). Better thus : if i. be the first co-efficient, M the greatest, and N the last, signs not considered, then all the roots, numerically speaking, lie between + L - and - • u + N 19. 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 +L) : L. And if L be the last co-efficient and u the greatest which has a different sign, no positive root is so small as L : (as + 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 coefficient, M the greatest which has a different sign, and if the first which has,a different sign be in the with place from the first term exclusive, or belong to the (m +1)th term; then no positive root is iso great as 1 + If k be the number of terms which elapse at the beginning before a change of sign, L the least of their coefficients, and m the greatest co-efficient of a different sign, any value of x which, being > 1, makes (rah - L - kr) 4.
positive, is greater than the greatest root : for instance, + x= L The following method is very convenient when the number of terms is large. Divide the whole expression into successive positive and negative lots, - C, D, . p, ry, r, s, &c., representing the last exponent of x in each lot. Divide - B, by xq , and ascertain a value of x, say A, which makes s,, - a, positive and equal to 1. Do the same with tx2 + c, - n, , which, for x = µ (perhaps no greater than A) becomes m. Repeat the process with nix' + re - r. , and so on to the end. The last value of .z used is greater than any root of the equation : and the first value of a., A, is very often the Last also.
21. If each co-efficient which differs in sign from the first term, be divided by the suns of all which precede and agree with the first term (the first term itself included), the greatest resulting fraction, increased by unity, is greater than any positive root of the equation.
22. Newton's method of finding a limit greater than the greatest positive root of any equation now merges in Fourier's theorem. It consists in finding a by inspection and trial, so that taa, 4:Ca, &c., shall all be positive.
23. Any mode of ascertaining a limit greater than the greatest posi tive root of an equation may be thus treated. Apply it to the reci procal equation (f 17), and the reciprocal of the result attained is less than the least positive root of the original. Apply both to the equa tion of roots with signs changed, and the results give limits for the negative roots of the original.
24. A celebrated mode of examining the roots of equations, hut too complicated for ordinary use, consists in forming the equation whose roots are the squares of the differences of the roots of the original. Any quantity being found less than the least positive root of this new equation, its square root is less than the difference of any two roots of the original. If such a quantity could be readily found, the theoretical imperfection of Fourier's theorem would be greatly diminished, and, practically speaking, much advantage would be gained in numerical solution. What is wanted to add to both Fourier's and Horner's method, is a ready mode of finding out when two roots are nearly equal.