Page:Scientific Papers of Josiah Willard Gibbs.djvu/288

252 infinitesimals of the higher orders. By substitution of these values, the condition of stability will reduce to the form or That is, the sum of the products of the volumes of the masses by their pressures, diminished by the sum of the products of the areas of the surfaces of discontinuity by their tensions, must be a maximum.

This is a purely geometrical condition, since the pressures and tensions are constant. This condition is of interest, because it is always sufficient for stability with reference to motion of surfaces of discontinuity. For any system may be reduced to the kind described by putting certain parts of the system in communication (by means of fine tubes if necessary) with large masses of the proper temperatures and potentials. This may be done without introducing any new movable surfaces of discontinuity. The condition (549) when applied to the altered system is therefore the same as when applied to the original system. But it is sufficient for the stability of the altered system, and therefore sufficient for its stability if we diminish its freedom by breaking the connection between the original system and the additional parts, and therefore sufficient for the stability of the original system.

The study of surfaces of discontinuity throws considerable light upon the subject of the stability of such homogeneous fluid masses as have a less pressure than others formed of the same components (or some of them) and having the same temperature and the same potentials for their actual components. In considering this subject, we must first of all inquire how far our method of treating surfaces of discontinuity is applicable to cases in which the radii of curvature of the surfaces are of insensible magnitude. That it should not be applied to such cases without limitation is evident from the consideration that we have neglected the term $$\tfrac{1}{2}(C{1} - C_{2}) \delta(c_{1} - c_{2})$$ in equation (494) on account of the magnitude of the radii of curvature compared with the thickness of the non-homogeneous film. (See page 228.) When, however, only spherical masses are considered, this term will always disappear, since $$C_{1}$$ and $$C_{2}$$ will necessarily be equal.