Part I

." _Quarterly Journal of Mathematics_, No. 1, April 1855; reprinted in Phil. Mag., January 1878, and in _Math. and Phys. Papers_, vol. i. pp. 291, seq.). Thus the striking new advance contained in the more modern work of J. Willard Gibbs (1875-1877) and of Helmholtz (1882) was rather the sustained general application of these ideas to chemical systems, such as the galvanic cell and dissociating gaseous systems, and in general fashion to heterogeneous concomitant phases. The fundamental paper of Kelvin connecting the electromotive force of the cell with the energy of chemical transformation is of date 1851, some years before the distinction between free energy and total energy had definitely crystallized out; and, possibly satisfied with the approximate exactness of his imperfect formula when applied to a Daniell's cell (_infra_), and deterred by absence of experimental data, he did not return to the subject. In 1852 he briefly announced (_Proc. Roy. Soc. Edin._) the principle of the dissipation of mechanical (or available) energy, including the necessity of compensation elsewhere when restoration occurs, in the form that "any restoration of mechanical energy, without more than an equivalent of dissipation, is impossible"--probably even in vital activity; but a sufficient specification of available energy (cf. _infra_) was not then developed. In the paper above referred to, where this was done, and illustrated by full application to solid elastic systems, the total energy is represented by c and is named "the intrinsic energy," the energy taken in during an isothermal transformation is represented by e, of which H is taken in as heat, while the remainder, the change of free (or mechanical or available) energy of the system is the unnamed quantity denoted by the symbol w, which is "the work done by the applied forces" at uniform temperature. It is pointed out that it is w and not e that is the potential energy-function for isothermal change, of which the form can be determined directly by dynamical and physical experiment, and from which alone the criteria of equilibrium and stress are to be derived--simply for the reason that for all _reversible_ paths at constant temperature between the same terminal configurations, there must, by Carnot's principle, be the same gain or loss of heat. And a system of formulae are given (5) to (11)--_Ex. gr._ e = w - t(dw/dt) + J [int]s dt for finding the total energy e for any temperature t when w and the thermal capacity s of the system, in a standard state, have thus been ascertained, and another for establishing connexion between the form of w for one temperature and its form for adjacent temperatures--which are identical with those developed by Helmholtz long afterwards, in 1882, except that the entropy appears only as an unnamed integral. The progress of physical science is formally identified with the exploration of this function w for physical systems, with continually increasing exactness and range--except where pure kinetic considerations prevail, in which cases the wider Hamiltonian dynamical formulation is fundamental. Another aspect of the matter will be developed below.

A somewhat different procedure, in terms of entropy as fundamental, has been adopted and developed by Planck. In an isolated system the trend of change must be in the direction which increases the entropy [phi], by Clausius' form of the principle. But in experiment it is a system at constant temperature rather than an adiabatic one that usually is involved; this can be attained formally by including in the isolated system (cf. _infra_) a source of heat at that temperature and of unlimited capacity, when the energy of the original system increases by [delta]E this source must give up heat of amount [delta]E, and its entropy therefore diminishes [delta]E/T. Thus for the original system maintained at constant temperature T it is [delta][phi] - [delta]E/T that must always be positive in spontaneous change, which is the same criterion as was reached above. Reference may also be made to H.A. Lorentz's _Collected Scientific Papers_,