25. Lagrange's mode of approximation is as follows :—Having found that a root of an equation lies between the integers a and a + 1, diminish all the roots of that equation by a, and take the reciprocal equation to the result. Find a root of the last lying between the integers b and 6+1, diminish all the roots by b, and take the reciprocal equation of the result. Find a root of this last between c and c+ 1, and proceed in the same way. Then the continued fraction 1 1 I — a + 6 + c is a root of the original. The details of the work are much abridged by use of llorneen process.
26. When an equation has equal roots, those roots can be found by an equation depending entirely on the different sets of equal roots. If cp.z have et roots equal to a, has m-1 of them, 4"x has ra-2 of them, and so on ; finally, has one of them. If then qs and o'x be found to have a common measure, every root of that common measure enters in cpx one time more than in the common measure itself.
27. When an equation has an integer root, which must be one of the divisors of the last co-efficient, it may be discovered by successive trial, as follows : —Suppose XS+ e+ a, s+ being integers. Let k be a divisor of a,, and let a, : 1:=1, an integer.
Then if k be a root, we have a,k + + 0, and a, + 1 is divisible by k, giving m, an integer. If once a,k + a + in= 0, and a, +m divided by k gives an integer, say O. Hence +a ft= 0, and a, + n divided by k gives — If all these courlitions be fulfilled, k is a root. All the divisors of a, being tried in this manner, settle the question of the integer !note entirely.
28. If the coefficients of an equation read backwards and forwards be the =IOC, both in sign and magnitude, every root has its reciprocal also among the roots. By reducing it to the form 1 1 p + 2 + + r + . . . = 0 which can always be clone by division, when the dimension is even, and assuming y= z+ an equation of the 2nth degree can be re duced to one of the nth and sr quadratics. But when the dimension is odd, either —1 or +I must be a root, and the equation can be depressed to an even degree by division by z+1 or z-1.
The student who is acquainted with the preceding results, namely, such as are either stated or referred to in this article, will find 310 difficulty either in reading on the history of this subject, or in its application. It is peculiarly a subject on which selection should be m.vlo for the beginner.