EQUILIBEIUM OF HETEROGENEOUS SUBSTANCES. 265 



potentials, and render the equations which involve them less fitted to 

 give a clear idea of physical relations. 



Now the fundamental equation of each of the homogeneous masses 

 which are separated by any surface of discontinuity affords a relation 

 between the pressure in that mass and the temperature and potentials. 

 We are therefore able to eliminate one or two potentials from the 

 fundamental equation of a surface by introducing the pressures in 

 the adjacent masses. Again, when one of these masses is a gas- 

 mixture which satisfies Dal ton's law as given on page 155, the 

 potential for each simple gas may be expressed in terms of the tem- 

 perature and the partial pressure belonging to that gas. By the 

 introduction of these partial pressures we may eliminate as many 

 potentials from the fundamental equation of the surface as there are 

 simple gases in the gas-mixture. 



An equation obtained by such substitutions may be regarded as a 

 fundamental equation for the surface of discontinuity to which it 

 relates, for when the fundamental equations of the adjacent masses 

 are known, the equation in question is evidently equivalent to an 

 equation between the tension, temperature, and potentials, and we 

 must regard the knowledge of the properties of the adjacent masses 

 as an indispensable preliminary, or an essential part, of a complete 

 knowledge of any surface of discontinuity. It is evident, however, 

 that from these fundamental equations involving pressures instead 

 of potentials we cannot obtain by differentiation (without the use of 

 the fundamental equations of the homogeneous masses) precisely the 

 same relations as by the differentiation of the equations between the 

 tensions, temperatures, and potentials. It will be interesting to 

 inquire, at least in the more important cases, what relations may be 

 obtained by differentiation from the fundamental equations just 

 described alone. 



If there is but one component, the fundamental equations of the 

 two homogeneous masses afford one relation more than is necessary 

 for the elimination of the potential. It may be convenient to regard 

 the tension as a function of the temperature and the difference of the 

 pressures. Now we have by (508) and (98) 



do- 

 d(p' -p") = ( 

 Hence we derive the equation 



p"), (578) 



which indicates the differential coefficients of o- with respect to t and 

 p' p". For surfaces which may be regarded as nearly plane, it is 



