An Elementary Treatise on the Theory of Equations: With a Collection of ExamplesMacmillan and Company, 1875 - 328 Seiten |
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a₁ algebraical approximation b₁ b₂ biquadratic equation c₁ c₂ changes of sign coefficients column contrary signs cubic equation d₁ denote derived function determinant divide divisible double permanences equa equal roots equation f(x equation f(x)=0 equation ƒ equation x³ example expression factors Fourier's functions given equation greater greatest common measure Hence homogeneous function identity imaginary roots integer method multiply negative root Newton's Newton's method nth degree nth roots number of changes obtain odd number original equation P₁ P₂ positive roots preceding Article proposed equation quadratic elements quotient qx² real roots reciprocal equation result rule of signs S₁ shew shewn solution solve the equation square Sturm's functions Sturm's theorem substitute suffixes superior limit suppose symmetrical function tion transformed equation unity unknown quantities vanish zero
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Seite 61 - ... any of the positive roots. For transform the proposed equation into one whose roots are the reciprocals of the roots of the proposed equation, and then the reciprocal of the superior limit of the positive roots of the transformed equation will be an inferior limit of the positive roots of the proposed equation. Thus suppose the proposed equation to be put — for x, and multiply by y...
Seite 86 - Thus an equation is a reciprocal equation when the coefficients of terms equidistant from the beginning and end are equal in magnitude and...
Seite 113 - L , a the equation in e should be of the sixth degree. But as the sum of the four roots of the biquadratic equation is zero by Art. 45, the sum of any two roots is equal in magnitude and opposite in sign to the sum of the remaining two roots ; and thus we see the reason why the equation in e only involves even powers of e, so that the values of e3 can be found by the solution of a cubic equation. We may observe that when we have found e...
Seite 29 - The coefficient pt of the fourth term with its sign changed is equal to the sum of the products of the roots taken three...
Seite 79 - Now, the annihilation of the second, third, and fourth terms of an equation may be made to depend upon the solution of a quadratic and a cubic equation between the same unknown quantities.! If the question were to be treated in this way, the method of Mr. Jerrard would require us to assume for y an expression, in terms of x, consisting of five terms, while mine would require an expression containing...
Seite 57 - Let f(x) = 0 be the equation ; suppose it of the w'h degree. Let p be the numerically greatest negative coefficient which occurs in f(x). Then if such a value be found for x that f(x) is positive for that value of x and for all greater values, that value is a superior limit of the positive roots of the equation f(x) = 0...
Seite 29 - Thus generally if pr denote as usual the coefficient of x"~r in the equation, (—1)^ = ^6 sum of the products of every r of the roots. 46. It might appear perhaps that the relations given in the preceding Article would enable us to find the roots of any proposed equation ; for they supply equations involving the roots, and the number of these equations is the same as the number of the roots, so that it might be supposed practicable to eliminate all the roots but one and thus to determine that root....
Seite 295 - B C D D A B C C D A B B C D A ment appears more than once either in the vertical or horizontal series. The Rothamsted plan is to put all the possible arrangements on cards in a hat and by drawing one out thus to fix the arrangement used by chance. The proper control of field experiments in which relatively small differences of...
Seite 45 - VARIATIONS of sign, nor the number of negative roots greater than the number of PERMANENCES. 325. Consequence. When the roots of an equation are all real, the number of positive roots is equal to the number of variations, and the number of negative roots is equal to the number of permanences. For, let m denote the degree of the equation, n the number of variations of the signs, p the number of permanences ; we shall have m=n+p. Moreover, let n' denote the number of positive roots, and p' the number...