The following Memoir is composed of two separate investigations, each of them having a general reference to the Theory of Determinants, but otherwise perfectly unconnected. The name of “ Determinants” or “ Resultants” has been given, as is well known, to the functions which equated to zero express the result of the elimination of any number of variables from as many linear equations, without constant terms. But the same functions occur in the resolution of a system of linear equations, in the general problem of elimination between algebraic equations, and particular cases of them in algebraic geometry, in the theory of numbers, and, in short, in almost every part of mathematics. They have accordingly been a subject of very considerable attention with analysts. Occurring, apparently for the first time, in Cramer's Introduction à l'Analyse des Lignes Courbes, 1750: they are afterwards met with in a Memoir On Elimination, by Bezout, Memoires de l'Académie, 1764; in two Memoirs by Laplace and Vandermonde in the same collection, 1774; in Bezout's Thérie générale des Equations algebriques [1779]; in Memoirs by Binet, Journal de l’École Poly technique, vol. ix. [1813]; by Cauchy, ditto, vol. x. [1815]; by Jacobi, Crelle's Journal, vol. xxn. [1841]; Lebesgue, Liouville, [vol. II. 1837], &c. The Memoirs of Cauchy and Jacobi contain the greatest part of their known properties, and may be considered as constituting the general theory of the subject. In the first part of the present paper, I consider the properties of certain derivational functions of a quantity U, linear in two separate sets of variables (by the term “ Derivational Function, ” I would propose to denote those functions, the nature of which depends upon the form of…