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‎Algebraic Researches, containing a disquisition on Newton's Rule for the Discovery of Imaginary Roots, and an allied Rule applicable to a particular class of Equations, together with a complete invariantive determination of the character of the Roots ...‎

‎(London, Taylor and Francis, 1865). 4to. No wrappers as extracted from ""Philosophical Transactions"", Vol. 154 - Part III, pp. 579-666 and 2 lithographed plates. Some faint brownspots to theplates. Otherwise clean and fine.‎

‎First printing of a main paper of the founder (together with Cayley) of the theory of algebraic invariants, in which Sylvester investigates the roots of quintic equations and the proof of Newton's rule.""Another problem of great importence investigated in two long memoirs of 1853 and 1864 (the paper offered) concerns the nature of the roots of quintic equation. Sylvester took the functions of the coefficients that serve to decide the reality of the roots, and treated them as coordinates of a point in n-dimensional space. A point is or is not ""facultative"" according to whether there correspons, or fail to correspond, an equation with real coefficients. The character of the roots depends on the bounding surface or surfaces of the facultative regions, and on a single surface depending on the ""discriminant.""(DSB XIII, p. 219).‎