Page:Scientific Papers of Josiah Willard Gibbs.djvu/292

256 required for the formation of such a globule of a certain size (viz., that which would be in equilibrium with the surrounding mass), will always be positive. Nor can smaller globules be formed, for they can neither be in equilibrium with the surrounding mass, being too small, nor grow to the size of that to which $$W$$ relates. If, however, by any external agency such a globular mass (of the size necessary for equilibrium) were formed, the equilibrium has already (page 243) been shown to be unstable, and with the least excess in size, the interior mass would tend to increase without limit except that depending on the magnitude of the exterior mass. We may therefore regard the quantity W as affording a kind of measure of the stability of the phase to which $$p''$$ relates. In equation (557) the value of $$W$$ is given in terms of $$\sigma$$ and $$p' - p''$$. If the three fundamental equations which give $$\sigma, p'$$, and $$p''$$ in terms of the temperature and the potentials were known, we might regard the stability $$(W)$$ as known in terms of the same variables. It will be observed that when $$p' = p''$$ the value of $$W$$ is infinite. If $$p' - p$$ increases without greater changes of the phases than are necessary for such increase, $$W$$ will vary at first very nearly inversely as the square of $$p' - p$$. If $$p' - p''$$ continues to increase, it may perhaps occur that $$W$$ reaches the value zero; but until this occurs the phase is certainly stable with respect to the kind of change considered. Another kind of change is conceivable, which initially is small in degree but may be great in its extent in space. Stability in this respect or stability in respect to continuous changes of phase has already been discussed (see page 105), and its limits determined. These limits depend entirely upon the fundamental equation of the homogeneous mass of which the stability is in question. But with respect to the kind of changes here considered, which are initially small in extent but great in degree, it does not appear how we can fix the limits of stability with the same precision. But it is safe to say that if there is such a limit it must be at or beyond the limit at which $$\sigma$$ vanishes. This latter limit is determined entirely by the fundamental equation of the surface of discontinuity between the phase of which the stability is in question and that of which the possible formation is in question. We have already seen that when $$\sigma$$ vanishes, the radius of the dividing surface and the work $$W$$ vanish with it. If the fault in the homogeneity of the mass vanishes at the same time (it evidently cannot vanish sooner), the phase becomes unstable at this limit. But if the fault in the homogeneity of the physical mass does not vanish with $$r, \sigma$$ or and $$W$$,—and no sufficient reason appears why this should not be considered as the general case,—although the amount of work necessary to upset the equilibrium of the phase is infinitesimal, this is not enough to make the phase unstable.