228 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



is evidently consistent with the suppositions made on page 219 with 

 regard to this surface. 



We may therefore cancel the term 



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

 C l must necessarily be equal to G 2 , when c^ c^, 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 approximately plane. Therefore, without 

 danger of sensible error, we may also cancel the term 



Equation (494) is thus reduced to the form 



Se s = tSn 8 + o-Ss + fjL 1 S'm% + iuL 2 S>m% + etc. (497) 



We may regard this as the complete value of Se s , for all reversible 

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

 when 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 

 discontinuity analogous to that expressed by equation (12) with 

 regard to homogeneous masses. From the two equations may be 

 directly deduced the conditions of equilibrium of heterogeneous 

 masses in contact, subject or not to the action of gravity, without 

 disregard of the influence of the surfaces of discontinuity. The 

 general problem, including the action of gravity, we shall take up 

 hereafter ; at present we shall only consider, as hitherto, a small part 

 of a surface of discontinuity 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, entropy, 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', jy', etc., e", r\ ', etc. Then the total energy will 

 be e 8 + e' -f e", and the general condition of internal equilibrium will be 



that 



^0, (498) 



