Strenuous efforts have been put forth by mathematicians to discover theorems by which the exact nuMber of real and of complex roots of equations with real coefficients can always be determined. The most noted result of these efforts is the theorem of J. C. F. Sturm, dis covered in 1829. Sturm's theorem tells the number of complex roots, and the number of real roots within a g:ven interval, with unfailing certainty; but it labors under the disadvantage of being laborious in its application. Hence it is commonly used only when the simpler methods fail to give the wanted information. We state the theorem for the spec:al case when f(x) —0 has no equal roots. Let f' Cr) be the first derived function of f(x). (See CALcutus). Then proceed with the process of finding, by division, the highest common factor of f(x) and r(x), with this modification, that the sign of each remainder he changed before it is used as a divisor. Continue the process until a re mainder is reached which does not contain x, and change the sign of that also. The functions f(x), 1(x), together with the several remain ders with their signs changed. viz.. fi(x), Mx), . . (x), are called °Sturm's functions.° Sturm's theorem is as follows: If f(x)=0 has no equa; roots, let any two real quantities a and b be substituted for x in Sturm's tions, then the difference between the number of variations of sign in the series when a is substituted for x and the number when b is substituted for x expresses the number of real roots of f(x)==0 bettveen a and b. To make this clearer, take f(x)= x' — x' —10x + 1, then f(x)---'3x* —2x —10, fa(x) --62x + 1, fi(x) =38,313. For the indicated values of x the signs of the Sturmian functions are as follows: Since x=c'D gives no variations and x-- 02 gives three variations, there are three real roots between 02 and — co. Hence there are no complex roots. The real roots lie between 3 and 4, 0 and 1, —2 and —3.
Transformations of Equations.— The study of the properties of an equation is frequently facilitated by the transformation of the given equation into a new one whose roots (coeffi cients) bear a given relation to the roots (co efficients) of the original equation. Thus, in applying Descartes' Rule to negative roots we transformed the equation into another whose roots were numencally the same, but differed in sign. If the roots of the new equation are to be m times those of the one given, we place y=mx and substitute On for x. For instance, if the roots of the transformed equation are to be 10 times those in y2 2y x3--x2-2x+ 5=0, we get — +5=-0, 1000 100 10 or y*-10?-200y+5000=0. The result is tained more easily by the rule: MuLiply the second term by m, the third by m', and so on.
If the roots of the new equation are to be the reciprocals of the roots of the old we write 1 x —. A more important transformation is the one of diminishing the roots by a given number h. We have here y=x—k. Substituting y+is for x in aftv -Faixn-2-Fasxm-2-1- -Fan=--0, we obtain cs.(y+h)s-Fai(Y±h)n-i +ch(Y±h)*--2+ ... +a =3.
Expanding the binomials and collecting like terms, we obtain, let us suppose, Awe + A ..=0.
Writing x—h for y we get Ao(x—h)n + Ai(x—h)L-1 +...+An-i(x—h)+A.=0, which differs from the original equation merely in form. This new fortn suggests an easy way for carrying mit the Rental compntation. Dividing the left member by x—k, the re mainder obtained is seen to be A., the abso lute term. Dividing the quotient thus obtained by x—h, the remainder is As-i. By repeating this process the remaining coefficients of the required equation are secured. The process, called *synthetic division"is very convenient S in this transformation. Suppose we desire to transform 1-4+8x.—x-F6=0 into another in which the second term is wanting. The sum of the roots is — 8 hence, to cause e to disappear, we must increase each root by 2 (i.e., dimin ish by —2). Dividing successively by x+2 we obtain the coefficients —40. 63. —24. 0, 1. and the required equation is x'-24.e+63x--40=0.
The transformations thus far considered are all special cases of the so-called homographic or projective transformation in which y= ax-1-b cx+d" a b, c, II being constants. Thus, ifa=d=1 and we have the preceding trans formation. The homographic transformation is of interest in geometry, in the study of homo graphic ranges of points. The most general rational algebraic transformation of the roots of an equation f(x)3 of the nth degree can always be reduced to an integral transformation of a degree not higher than the (n—l)th, and can, therefore, be represented by the relation y= + dixs . . .
This last is known as the °Tschirnhausen trans formation,* by which Tschirnhausen in 1683 hoped to be able to reduce the general equation of the nth degree to the binomial form xn—a=0, which is always solvable. But this transformation to the binomial form can be effected only for general equations that are lower than the fifth degree.
Solution of Equat:ons.—This subject re solves itself into two quite distinct parts: (1) The solution of numerical equations (i.e., equations whose coefficients are given numbers) by some method of approximation to the exact value of the roots; (2) the solution of equa tions, whose coefficients are either given num bers or letters, by operations which will give the accurate values of the roots, expressed in terms of the coefficients,— such expressions to involve no other processes than addition, sub traction, multiplication, division and the ex traction of roots. The former is called a solu tion by approximation, the second is called the algebraic solution of equations. In the former each root may be found separately, in the lat ter a general expression is obtained which rep resents all of the roots indifferently. The for mer is of importance to the practical computor, the latter is of special interest to the pure mathematician. The solution by approximation can be effected for equations of any degree; the algebraic solution is impossible for general equations of the fifth or of higher degrees. See EQUATFMS, GALOTS' THEORY OF.