On a Theory of the Syzygetic relations of two rational integral functions, comprising an application to the Theory of Sturm's Functions, and that of the greatest Algebraic Common Measure. Received and Read June 16, 1853.

(London, Richard Taylor and William Francis, 1853) 4to. No wrappers as extracted from "Philosophical Transactions" 1853, Vol. 143 - Part III. With the titlepage to Vol.153 - Part III. a. pp. 397-406. Clean and fine. First printing of a main paper of the founder (together with Cayley) of the theory of algebraic invariants. This theory involved a more general theory of "covariants", more concretely, given a binary form of a particular degree, Sylvester and Cayley devised techniques both for explicitly finding invariants and covariants of that form and for determining algebraic relations, or "Syzygetic", between them. Sylvester tackled these problems in the paper offered, and proving among other results "Sylvester's Law of Inertia."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).…