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



mixture which satisfies Dalton's law as given on page 215, 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 wdll 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 tlie 

 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) 



(Iff =. — 7/s dt — rdf.{ , , 

 d{p'-p") = (Vv'-O dt + (/-;/') dju,. 

 Hence we derive the equation 



dff = - (^Th - -,^^7/ iVv' - '/v")) dt ^ ^r^—r, d {p -p"), (578) 



which indicates the differential coefficients of ff with respect to t and 

 p' — p". For surfaces w^hich may be regarded as nearly plane, it is 



evident that —. j. represents the distance from the surface of ten- 



y -y 



sion to a dividing surface located so as to make the superficial 

 density of the single component vanish, (being positive, when the 



