230 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



which may be applied to any portion of any surface of discontinuity 

 (in equilibrium) which is of the same nature throughout, or through- 

 out which the values of t, a; fJ. l) /m 2 , etc., are constant. 



If we differentiate this equation, regarding all the quantities as 

 variable, and compare the result with (501), we obtain 



rf 1 dt + sdv-\- m?cfy/ 1 -fmfcZya 2 + etc. = 0. (503) 



If we denote the superficial densities of energy, of entropy, and of 

 the several component substances (see page 224) by e s , ij S) T lt T 2 , etc., 

 we have 



g = ^, % =3-, (504) 



]?! = , r 2 = , etc., (505) 



and the preceding equations may be reduced to the form 



(506) 

 + etc., (507) 



da- = J] 8 dt r i c? y w 1 T%djUL 2 etc. (508) 



Now the contact of the two homogeneous masses does not impose 

 any restriction upon the variations of phase of either, except that 

 the temperature and the potentials for actual components shall have 

 the same value in both. {See (482)-(484) and (500).} For however 

 the values of the pressures in the homogeneous masses may vary (on 

 account of arbitrary variations of the temperature and potentials), 

 and however the superficial tension may vary, equation (500) may 

 always be satisfied by giving the proper curvature to the surface of 

 tension, so long, at least, as the difference of pressures is not great. 

 Moreover, if any of the potentials JUL I , ju. 2) etc., relate to substances 

 which are found only at the surface of discontinuity, their values 

 may be varied by varying the superficial densities of those sub- 

 stances. The values of t, JUL I} JUL Z) etc., are therefore independently 

 variable, and it appears from equation (508) that o- is a function of 

 these quantities. If the form of this function is known, we may 

 derive from it by differentiation n+I equations (n denoting the total 

 number of component substances) giving the values of ?/ s , I\, F 2 , 

 etc., in terms of the variables just mentioned. This will give us, 

 with (507), 7i+3 independent equations between the 2^ + 4 quantities 

 which occur in that equation. These are all that exist, since n + l 

 of these quantities are independently variable. Or, we may consider 

 that we have n+3 independent equations between the 2n+5 quan- 

 tities occurring in equation (502), of which n + 2 are independently 

 variable. 



