•7. ]V. Gibhs — Equllihrimn of Heterogeneous Substances. 20V 



in phase, is due to the fact that we have made the elements of volume 

 variable in position and size, while the matter initially contained in 

 these elements is not supposed to be confined to them. Now, as the 

 different components may move in different directions when the 

 state of the system varies, it is evidently imi)ossible to define the 

 elements of volume so as always to include the same matter ; we 

 must, therefore, suppose the matter contained in the elements of 

 volume to vary ; and therefore it would be allowable to make these 

 elements fixed in space. If the given mass has no surfaces of discon- 

 tinuity, this would be much the simplest plan. But if there are any 

 surfaces of discontinuity, it will be possible for the state of the given 

 mass to vary, not only by infinitesimal changes of phase in the fixed 

 elements of volume, but also by movements of the surfaces of discon- 

 tinuity. It would therefore be necessary to add to our general con- 

 dition of equilibrium terms relating to discontinuous changes in the 

 elements of volume about these surfaces, — a necessity which is 

 avoided if we consider these elements movable, as we can then sup- 

 pose that each element remains always on the same side of the surface 

 of discontinuity. 



Method of treating the preceding jyrohlem^imiiMch the elements of 

 volume are regarded as fixed. 



It may be interesting to see in detail how the particular conditions 

 of equilibrium may be obtained if we regard the elements of volume 

 as fixed in position and size, and consider the possibility of finite as 

 well as infinitesimal changes of phase in each element of volume. If 

 we use the character A to denote the differences determined by such 

 finite differences of phase, we may express the variation of the intrin- 

 sic energy of tlie whole mass in the form 



fSBe + fABe, (237) 



in which the first integral extends over all the elements which are 

 infinitesimally varied, and the second over all those which experience 

 a finite variation. We may regard both integrals as extending 

 throughout the whole mass, but their values will be zero except for 

 the parts mentioned. 



If we do not wish to limit ourselves to the consideration of masses 

 so small that the force of gravity can be regarded as constant 

 in direction and in intensity, we may use T to denote the potential of 

 the force of gravity, and express the variation of the part of the 

 energy which is due to gravity in the form 



-y r 6 Dm -fT ADm. (238) 



