...The same conditions maybe found by equating to zero the expression, in terms of the coefficients, for the product of the squares of the differences of the roots of the cubic equation, which has been given in a very remarkable form, as the sum of seven squares, by...

...expression determinable for an equation of the next inferior degree. Let qm, g^ denote, respectively, the product of the squares of the differences of the roots of f(x) = 0, and of r2 + &c. = 0 . . . (1), fore qm, being an integral function of one of the roots of...

...the theory of equations, that the left-hand member of the equation last written is proportional to the product of the squares of the differences of the roots of the equation in k ; and that when it is positive the roots of the equation in k are all real, but that...

...the equation x 3 — вx 2 -f Hz — 6 = 0 are in Arithmetical Progression ; find them. T. Prove that the product of the squares of the differences of the roots of an equation may be exhibited as a determinant. 8. If two rows or two columns of a determinant are identical,...

...quadratic is positive when the roots are real, and negative when they are imaginary. It follows then that the product of the squares of the differences of the roots of any quantic is positive when it has an even number of pairs of imaginary roots, and negative when it...

...discriminant of the function in a: is, in all cases, the absolute term in the equation in z. ¿i. If Д be the product of the squares of the differences of the roots of x3 - рхг + qx - f * o, find an expression in terms of the roots, for — , by sob1 ing from three...

...discriminant of the function in * is, in all cases, thu absolute term in the equation in z. 5i. If A be the product of the squares of the differences of the roots of <fA find an expression in terms of the roots, for — , by solving from three equations of the form...

...; on the detection and precipitation of phosphoric acid by ammonic molybdate, by AN M 'Alpine ; on the product of the squares of the differences of the roots of a cubic equation, by Prof. Young ; and on a new genus and species of sponge, by Prof. EP Wright. THE...

...locus of N is the evolute of the locus of T. 6426. (By 0. LEUDESDORF, MA)— Prove that, if «be odd, the product of the squares of the differences of the roots of the equation nl + (n — l)n-1 = 0 is 2»" (n —1) (»-Ч«. Solution by the Bev. D. THOMAS, MA ;...

............................................................. 43 6426. (С. Lendesdorf, MA)— Prove that, if и Ъе odd, the product of the squares of the differences of the roots of the equation + (nl)»-= 0 is 2n" (nl)t"-1)' ............. 40 6436. (D. Edwardes.) — If a quadrilateral...