Page:Scientific Papers of Josiah Willard Gibbs.djvu/276

To make this incapable of a negative value, we must have

$\mu _{g}''=\mu _{g}',$⁠unless⁠$m_{g}^{S''}=0,$ $\mu _{h}''=\mu _{h}',$⁠unless⁠$m_{h}^{S''}=0.$

In virtue of these relations and by equation (502), the expression (519), i.e., (516), will reduce to

$\sigma ''s-\sigma 's,$

which will be positive or negative according as

$\sigma ''-\sigma '$

(520)

is positive or negative.

That is, if the tension of the film is less than that of any other film of the same components which can exist between the same homogeneous masses (which has therefore the same values of $t,\mu _{a},\mu _{b}$, etc.), and which moreover has the same values of the potentials $\mu _{g},\mu _{h}$, etc., so far as it contains the substances to which these relate, then the first film will be stable. But the film will be practically unstable, if any other such film has a less tension. (Compare the expression (141), by which the practical stability of homogeneous masses is tested.)

It is, however, evidently necessary for the stability of the surface of discontinuity with respect to deformation, that the value of the superficial tension should be positive. Moreover, since we have by (502) for the surface of discontinuity

$\epsilon ^{S}-t\eta ^{S}-\mu _{a}m_{a}^{S}-\mu _{b}m_{b}^{S}-{\text{etc.}}-\mu _{g}m_{g}^{S}-\mu _{h}m_{h}^{S}-{\text{etc.}}=\sigma s,$

and by (93) for the two homogeneous masses

$\epsilon '-t\eta '+pv'-\mu _{a}m_{a}'-\mu _{b}m_{b}'-{\text{etc.}}=0,$ $\epsilon ''-t\eta ''+pv''-\mu _{a}m_{a}''-\mu _{b}m_{b}''-{\text{etc.}}=0,$

if we denote by

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

the total energy, etc. of a composite mass consisting of two such homogeneous masses divided by such a surface of discontinuity, we shall have by addition of these equations

$\epsilon -t\eta +pv-\mu _{a}m_{a}-\mu _{b}m_{b}-{\text{etc.}}-\mu _{g}m_{g}-\mu _{h}m_{h}-{\text{etc.}}=\sigma s.$

Now if the value of $\sigma$ is negative, the value of the first member of this equation will decrease as $s$ increases, and may therefore be decreased by making the mass to consist of thin alternate strata of the two kinds of homogeneous masses which we are considering. There will be no limit to the decrease which is thus possible with a given value of $v$, so long as the equation is applicable, i.e., so long as the strata have the properties of similar bodies in mass. But it