390 J^. W. Gibbs — Jiquilibrima of Heterogeneous iiubstances. 

 We may therefore cancel the term 



in (494). In regard to the following term, it will be ohserved that 

 (7j must necessarily be equal to Cg, when c, = f'2, which is the case 

 when the surface of discontinuity is plane. Now on account of the 

 thinness of the non-homogeneous film, we may always regard it as 

 composed of parts which are ajiproximately plane. Therefore, with- 

 out danger of sensible error, we may also cancel the term 



Equation (494) is thus reduced to the form 



6t^ = t Sif + 6s+ 1.1^ 6m\ + yWg ^'^'1 + ^^c. (497) 



We may regard this as the complete value of dt^, for all reversible 

 variations in the state of the system supposed initially in equilibrium, 

 Avhen the dividing surface has its initial position determined in the 

 manner described. 



The above equation is of fundamental importance in the theory 

 of capillarity. It expresses a relation with regard to surfaces of dis- 

 continuity analogous to that expressed by equation (12) with regard 

 to homogeneous masses. From the two equations may be directly 

 deduced the conditions of eqiiilibrium of heterogeneous masses in con- 

 tact, subject or not to the action of gravity, without disregard of the 

 influence of the surfaces of discontinuity. The general problem, in- 

 cluding the action of gravity, we shall take up hereafter: at present 

 we shall only consider, as hitherto, a small part of a surface of dis- 

 continuity with a part of the homogeneous mass on either side, in 

 order to deduce the additional condition which may be found when 

 we take account of the motion of the dividing surface. 



We suppose as before that the mass especially considered is 

 bounded by a surface of which all that lies in the region of non- 

 homogeneity is such as may be traced by a moving normal to the 

 dividing surface. But instead of dividing the mass as before into 

 four parts, it will be sufficient to regard it as divided into two parts 

 by the dividing surface. The energy, entro])y, etc., of these parts, 

 estimated on the supposition that its nature (including density of 

 energy, etc.) is uniform quite up to the dividing surface, will be 

 denoted by e', ?/, etc., e", y/", etc. Then the total energy will be 

 f^-f-f'-f f", and the general condition of internal equilibrium will be 



that 



d6Htff'+(56"^0, (498) 



