Page:Scientific Papers of Josiah Willard Gibbs.djvu/331

$A,B,C$ meet is stable with respect to the formation of the fluid $D$. When $p_{D}$ has a greater value, if such a line can exist at all, it must be at least practically unstable, i.e., if only a very small mass of the fluid $D$ should be formed it would tend to increase.

Let us next consider the case in which the tensions of the new surfaces are too small to be represented as in figure 15. If the pressures and tensions are consistent with equilibrium for any very small value of $v_{D}$, the angles of each of the curvilinear triangles $adb,bdc,cda$ will be together less than two right angles, and the lines $ab,be,ca$ will be convex toward the mass $D$. For given values of the pressures and tensions, it will be easy to determine the magnitude of $v_{D}$. For the tensions will give the total curvatures (in degrees) of the lines $ab,bc,ca$; and the pressures will give the radii of curvature. These lines are thus completely determined. In order that $v_{D}$ shall be very small it is evidently necessary that $p_{D}$ shall be less than the other pressures. Yet if the tensions of the new surfaces are only a very little too small to be represented as in figure 15, $v_{D}$ may be quite small when the value of $p_{D}$ is only a little less than that given by equation (636). In any case, when the tensions of the new surfaces are too small to be represented as in figure 15, and $v_{D}$ is small, $W_{V}$ is negative, and the equilibrium of the mass $D$ is stable. Moreover, $W_{S}-W_{V}$, which represents the work necessary to form the mass D with its surfaces in place of the other masses and surfaces, is negative.

With respect to the stability of a line in which the surfaces ${\text{A-B, B-C, C-A}}$ meet, when the tensions of the new surfaces are too small to be represented as in figure 15, we first observe that when the pressures and tensions are such as to make $v_{D}$ moderately small but not so small as to be neglected (this will be when $p_{D}$ is somewhat smaller than the second member of (636),—more or less smaller according as the tensions differ more or less from such as are represented in figure 15), the equilibrium of such a line as that supposed (if it is capable of existing at all) is at least practically unstable. For greater values of $p_{D}$ (with the same values of the other pressures and the tensions) the same will be true. For somewhat smaller values of $p_{D}$, the mass of the phase $D$ which will be formed will be so small, that we may neglect this mass and regard the surfaces ${\text{A-B, B-C, C-A}}$ as meeting in a line in stable equilibrium. For still smaller values of $p_{D}$, we may likewise regard the surfaces ${\text{A-B, B-C, C-A}}$ as capable of meeting in stable equilibrium. It may be observed that when $v_{D}$, as determined by our equations, becomes quite insensible, the conception of a small mass $D$ having the properties deducible from our equations ceases to be accurate, since the matter in the vicinity of a line where these surfaces of discontinuity meet must be