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any rate not more prominent than the cognate fundamental discussions on convergency and functionality have become in pure mathematical analysis. A field in which these two types of approximation towards ideal exact procedure have something in common is sketched in the analysis of Appendix A: it is now recognized that, in a strictly rigorous presentation of the theory of gravitational and other agencies, it is necessary to inquire (at considerable length) how far a potential is a function that can legitimately be differentiated at a point inside the attracting mass: it is here explained, among other things, that in a physical problem the potential about which these mathematical discussions arise is not the potential of the actual molecular distribution of matter but that of an ideal smoothed-out or continuous distribution, in fact a mechanical^[1] representation which is, only in certain definite respects, equivalent to it. Refinements of this kind are no doubt foreign to an empirical formulation of mathematical physics, where the aim is merely a concise expression of the facts of observation; but it seems not unlikely that in the final theory of the transition from molecular to mechanical dynamics they will be of fundamental import.

The main general principle in this domain, which is considered in some aspects more at length in the memoirs above referred to, is that the mechanical and the molecular properties of a material system, which is not undergoing constitutive change, are independent of each other and not mutually involved: so that the mechanical interactions of a system can be developed independently of a knowledge of its molecular constitution. This principle may even be sufficiently deep-seated to have a bearing on the solution of the philosophical question of ultimate mechanical determinism.

energetics can supply an adequate clue to the course of physical and chemical change.

1. Throughout this Essay the term mechanical is used in antithesis to molecular: mechanics is the dynamics of matter in bulk, in contrast with molecular dynamics.