J. W. Gibhs — Equilibrium of Heterogeneous Substances. 421 



cannot vanish sooner,) tlie ])liase becomes unstable at this limit. 

 But if the fault in the homogeneity of the physical mass does not 

 vanish with r, and W, — and no sufficient reason appears why this 

 should not be considered as the general case, — although the amount 

 of Avork necessary to upset the equilibrium of the phase is infinitesi- 

 mal, this is not enough to make the phase unstable. It appears 

 therefoi-e that W is a somewhat one-sided measure of stability. 



It must be remembered in this connection that the fundamental 

 equation of a surface of discontinuity can hardly be regarded as 

 capable of experimental determination, except for plane surfaces, (see 

 j)p. 394, 395,) although the relation for spherical surfaces is in the 

 nature of things entirely determined, at least so far as the phases are 

 separately capable of existence. Yet the foregoing discussion yields 

 the following practical results. It has been shown that the real 

 stability of a phase extends in general beyond that limit (discussed 

 on pages 160, 161), which may be called the limit of practical stabil- 

 ity, at which the phase can exist in contact with another at a plane 

 surface, and a formula has been deduced to express the degree of 

 stability in such cases as measured by the amount of work necessary 

 to upset the equilibrium of the phase when supposed to extend indefi- 

 nitely in space. It has also been shown to be entirely consistent 

 with the principles established tliat this stability should have limits, 

 and the manner in which the general equations would accommodate 

 themselves to this case has been pointed out. 



By equation (553), wdnch may be written 



W= as - {})' - p") v', (559) 



we see that the work TF consists of two parts, of which one is always 

 positive, and is expressed by the product of the superficial tension 

 and the area of the surface of tension, and the other is always nega- 

 tive, and is numerically equal to the product of the difierence of pres- 

 sure by the volume of the interior mass. We may regard the first 

 part as expressing the work spent in forming the surface of tension, 

 and the second part the work gained in forming the interior mass.* 



* To make the physical significance of the above more clear, we may suppose the 

 two processes to be performed separately in the following manner. We may sup- 

 pose a large mass of the same phase as that which has the volume v' to exist 

 initially in the interior of the other. Of course, it must be surrounded by a resisting 

 envelop, on account of the difference of the pressures. We may, however, suppose 

 this envelop permeable to all the component substances, although not of such proper- 

 ties that a mass can form on the exterior like that within. We may allow the 



