EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 239 



increment of energy of the surface, the above condition expresses 

 that the increment of the total energy of the system is positive. 

 That we have only considered the possible formation of such films as 

 are capable of existing in equilibrium between the given homogeneous 

 masses can not invalidate the conclusion in regard to the stability of 

 the film, for in considering whether any state of the system will have 

 less energy than the given state, we need only consider the state of 

 least energy, which is necessarily one of equilibrium. 



If the expression (516) is capable of a negative value for an 

 infinitesimal change in the nature of the part of the film to which 

 the symbols relate, the film is obviously unstable. 



If the expression is capable of a negative value, but only for finite 

 and not for infinitesimal changes in the nature of this part of the 

 film, the film is practically unstable* i.e., if such a change were 

 made in a small part of the film, the disturbance would tend to 

 increase. But it might be necessary that the initial disturbance 

 should also have a finite magnitude in respect to the extent of 

 surface in which it occurs ; for we cannot suppose that the thermo- 

 dynamic relations of an infinitesimal part of a surface of discontinuity 

 are independent of the adjacent parts. On the other hand, the 

 changes which we have been considering are such that every part 

 of the film remains in equilibrium with the homogeneous masses 

 on each side ; and if the energy of the system can be diminished by 

 a finite change satisfying this condition, it may perhaps be capable 

 of diminution by an infinitesimal change which does not satisfy the 

 same condition. We must therefore leave it undetermined whether 

 the film, which in this case is practically unstable, is or is not 

 unstable in the strict mathematical sense of the term. 



Let us consider more particularly the condition of practical stability, 

 in which we need not distinguish between finite and infinitesimal 

 changes. To determine whether the expression (516) is capable of a 

 negative value, we need only consider the least value of which it is 

 capable. Let us write it in the fuller form 



e 8 " - e 8 ' - t(f - f) - f^ (mf - mf) - /* 6 (mf - mf) - etc. \ _, 



- Xr( m f - m ?') - /4Of - wf) - etc.,/ 



where the single and double accents distinguish the quantities which 

 relate to the first and second states of the film, the letters without 

 accents denoting those quantities which have the same value in both 

 states. The differential of this expression when the quantities distin- 

 guished by double accents are alone considered variable, and the area 

 of the surface is constant, will reduce by (501) to the form 



*With respect to the sense in which this term is used, compare page 79. 



