420 J. W. Gibbs — Eqidlibrmm of Heterogeneous Substances. 



Within those limits within whicli the method used has been justified, 

 the mass in question must be regarded as in strictness stable with 

 respect to the growth of a globule of the kind considered, since TF, 

 the work required for the formation of such a globule of a certain 

 size (viz., that which would be in equilibrium with the sm-rounding 

 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 40(3) 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 TF as affording a kind of measure of 

 the stability of the phase to which p" relates. In equation (55*7) the 

 value of TF is given in terms of <T and jw' —2^"- If the three funda- 

 mental equations which give C, ^j', and p" in terms of the tempera- 

 ture 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'=zj)" the value of TF is infinite. If p' —p" increases 

 without greater changes of the phases than are necessary for snch 

 increase, TF will vai-y at first very nearly inversely as the square of 

 p'—p". \S. p' —p" continues to increase, it may perhaps occur that 

 TFreaches 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 conthitious changes of phase has already been discussed 

 (see page 162), 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 tlie 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 whicli 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 o' vanislies, the radius of the divid- 

 ing surface and the Avork W vanish with it. If the fault in the 

 homogeneity of the mass vanishes at the same time, (it evidently 



