Elementary Theory of Equations

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J. Wiley & sons, Incorporated, 1914 - Equations, Theory of - 184 pages
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Page 33 - Article ; for the product of the squares ot the differences of all the roots is made up of the product of the squares of the differences of the roots of...
Page 134 - ... running from the upper left hand corner to the lower right hand corner of the symbol for the determinant.
Page 171 - These criteria were derived by a new transformation, namely the one which yields an equation whose roots are the squares of the differences of the roots of the given...
Page 140 - ... column is the sum of the products of the elements of the Tth row of the first determinant by the corresponding elements of the cth column of the second determinant.
Page 25 - Hence the modulus of the product of two complex numbers equals the product of their moduli, and the amplitude of the product equals the sum of their amplitudes.
Page 91 - Hence the coordinates of the various points located by the construction, and therefore also the length xi of the segment joining two of them, are found by a finite number of rational operations and extractions of real square roots, performed upon rational numbers and numbers obtained by earlier ones of these operations.
Page 27 - THEOREM s. 377. If the number of sides of a regular inscribed polygon is indefinitely increased, the apothem of the polygon approaches the radius of the circle as its limit. Given a regular polygon of n sides inscribed in the circle of radius OA, s being one side and a the apothem. To prove that a approaches r as a limit, if n is increased indefinitely. Proof. We know that a < r.
Page 102 - И 0. For, if it were zero, x — p would by (1) be a common factor of / and /i. We may now proceed as in the second case in 4. The third case requires a modified proof only when r is a multiple root. Let r be a root of multiplicity т, т Ш 2.
Page 90 - Another famous problem of antiquity was the construction of a cube whose volume shall be double that of a given cube.
Page 72 - I. The logarithm of a product equals the sum of the logarithms of the factors.

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