EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 257 



It appears therefore 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 

 pp. 231-233), 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 103-105), which may be called the limit of practical stability, 

 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 

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

 with the principles established that 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), which may be written 



W=<rs-(p'-p")v', (559) 



we see that the work W 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 

 negative, and is numerically equal to the product of the difference 

 of pressure 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.* Moreover, the second of these quantities, if we neglect its 



* 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 suppose 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 properties that a mass can form 

 on the exterior like that within. We may allow the envelop to yield to the internal 

 pressure until its contents are increased by v' without materially affecting its superficial 

 area. If this be done sufficiently slowly, the phase of the mass within will remain 

 constant. (See page 84.) A homogeneous mass of the volume v' and of the desired 

 phase has thus been produced, and the work gained is evidently (p 1 -p")v'. 



Let us suppose that a small aperture is now opened and closed in the envelop so as 

 to let out exactly the volume v' of the mass within, the envelop being pressed, inwards 

 in another place so as to diminish its contents by this amount. During the extrusion of 

 the drop and until the orifice is entirely closed, the surface of the drop must adhere to 

 the edge of the orifice, but not elsewhere to the outside surface of the envelop. The 

 work done in forming the surface of the drop will evidently be <rs or %(p' -p")tf. Of 

 G. I. R 



