Page:Scientific Papers of Josiah Willard Gibbs.djvu/277

may easily be shown (as in a similar case on pages 77, 78) that when the values of

$t,$$p,$$\mu _{a},$$\mu _{b},$etc.,$\mu _{g},$$\mu _{h},$etc.,

are regarded as fixed, being determined by the surface of discontinuity in question, and the values of

$\epsilon ,$$\eta ,$$m_{a},$$m_{b},$etc.,$m_{g},$$m_{h},$etc.,

are variable and may be determined by any body having the given volume $v$, the first member of this equation cannot have an infinite negative value, and must therefore have a least possible value, which will be negative, if any value is negative, that is, if $\sigma$ is negative.

The body determining $\epsilon ,\eta$, etc. which will give this least value to this expression will evidently be sensibly homogeneous. With respect to the formation of such a body, the system consisting of the two homogeneous masses and the surface of discontinuity with the negative tension is by (53) (see also page 79) at least practically unstable, if the surface of discontinuity is very large, so that it can afford the requisite material without sensible alteration of the values of the potentials. (This limitation disappears, if all the component substances are found in the homogeneous masses.) Therefore, in a system satisfying the conditions of practical stability with respect to the possible formation of all kinds of homogeneous masses, negative tensions of the surfaces of discontinuity are necessarily excluded.

Let us now consider the condition which we obtain by applying (516) to infinitesimal changes. The expression may be expanded as before to the form (519), and then reduced by equation (502) to the form

$s(\sigma ''-\sigma ')+m_{g}^{S''}(\mu _{g}''-\mu _{g}')+m_{h}^{S''}(\mu _{h}''-\mu _{h}')+{\text{etc.}}$

That the value of this expression shall be positive when the quantities are determined by two films which differ infinitely little is a necessary condition of the stability of the film to which the single accents relate. But if one film is stable, the other will in general be so too, and the distinction between the films with respect to stability is of importance only at the limits of stability. If all films for all values of $\mu _{g},\mu _{h}$, etc. are stable, or all within certain limits, it is evident that the value of the expression must be positive when the quantities are determined by any two infinitesimally different films within the same limits. For such collective determinations of stability the condition may be written

$-s\Delta \sigma -m_{g}^{S}\Delta \mu _{g}-m_{h}^{S}\Delta \mu _{h}-{\text{etc.}}>0,$

or

$\Delta \sigma <-\Gamma _{g}\Delta \mu _{g}-\Gamma _{h}\Delta \mu _{h}-{\text{etc.}}$

(521)

On comparison of this formula with (508), it appears that within the limits of stability the second and higher differential coefficients of the