§ 1. In a verbal communication to the Royal Society of Edinburgh in 1876, under the title “Thermodynamic Motivity,” I suggested the name motivity to express energy, whether thermal or of any other kind, available to generate velocity in molar matter, or to move molar matter against resisting force. By molar matter I mean matter as we know it, consisting, as we believe, of vast numbers of atoms or molecules. Only the title of this communi- cation was published in the Proceedings of tlie Royal Society. a short report of it was published in the Philosophical Magazine for May 1879, in which it was pointed out that a “very short and simple analytical method of setting forth the whole non-molecular theory of thermodynamics” might be founded on the consideration of Motivity and Energy as two functions of all the independent variables specifying a body, or a system of bodies, or some definite apparatus under consideration:—apparatus, as I shall call it for brevity, to include every case, even such as a single crystal. The object of the present communication is to carry out this proposal.

§ 2. Let the apparatus be given all at one temperature t. Denote by g1, g2, g3, etc, the other variables by which its condition is specified. These will in many cases be geometrical, specifying elements or co-ordinates, such as strain-components, expressing change of bulk or shape of a piece of crystal or other elastic solid under stress; or positions of pistons in a pneumatic apparatus; or area, or curvature, of a free liquid surface in an application to theory of capillary attraction; or positions of electrified bodies, or electrostatic capacities, in an electrostatic system.