Page:Scientific Papers of Josiah Willard Gibbs.djvu/301

potentials, and render the equations which involve them less fitted to give a clear idea of physical relations.

Now the fundamental equation of each of the homogeneous masses which are separated by any surface of discontinuity affords a relation between the pressure in that mass and the temperature and potentials. We are therefore able to eliminate one or two potentials from the fundamental equation of a surface by introducing the pressures in the adjacent masses. Again, when one of these masses is a gas-mixture which satisfies Dalton's law as given on page 155, the potential for each simple gas may be expressed in terms of the temperature and the partial pressure belonging to that gas. By the introduction of these partial pressures we may eliminate as many potentials from the fundamental equation of the surface as there are simple gases in the gas-mixture.

An equation obtained by such substitutions may be regarded as a fundamental equation for the surface of discontinuity to which it relates, for when the fundamental equations of the adjacent masses are known, the equation in question is evidently equivalent to an equation between the tension, temperature, and potentials, and we must regard the knowledge of the properties of the adjacent masses as an indispensable preliminary, or an essential part, of a complete knowledge of any surface of discontinuity. It is evident, however, that from these fundamental equations involving pressures instead of potentials we cannot obtain by differentiation (without the use of the fundamental equations of the homogeneous masses) precisely the same relations as by the differentiation of the equations between the tensions, temperatures, and potentials. It will be interesting to inquire, at least in the more important cases, what relations may be obtained by differentiation from the fundamental equations just described alone.

If there is but one component, the fundamental equations of the two homogeneous masses afford one relation more than is necessary for the elimination of the potential. It may be convenient to regard the tension as a function of the temperature and the difference of the pressures. Now we have by (508) and (98)

$d\sigma =-\eta _{S}dt-\Gamma d\mu _{1},$ $d(p'-p'')=(\eta _{V}'-\eta _{V}'')dt+(\gamma '-\gamma '')d\mu _{1}.$

Hence we derive the equation

$d\sigma =-\left(\eta _{S}-{\frac {\Gamma }{\gamma '-\gamma ''}}(\eta _{V}'-\eta _{V}'')\right)dt-{\frac {\Gamma }{\gamma '-\gamma ''}}(p'-p''),$

(578)

which indicates the differential coefficients of $\sigma$ with respect to $t$ and $p'-p''$. For surfaces which may be regarded as nearly plane, it is