EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 251 



The total quantity of the substance indicated by the suffix x is 



Making this constant, we have 



** (546) 



The condition of equilibrium is thus reduced to the form 



da \ 2 



,dy' 

 -^ 



fj Q fl 2o 



where -j f and -^-7 are to be determined from the form of the surface 

 dv dv 



of tension by purely geometrical considerations, and the other differ- 

 ential coefficients are to be determined from the fundamental equations 

 of the homogeneous masses and the surface of discontinuity. Condition 

 (540) may be easily deduced from this as a particular case. 



The condition of stability with reference to motion of surfaces of 

 discontinuity admits of a very simple expression when we can treat 

 the temperature and potentials as constant. This will be the case 

 when one or more of the homogeneous masses, containing together 

 all the component substances, may be considered as indefinitely large, 

 the surfaces of discontinuity being finite. For if we write 2Ae for 

 the sum of the variations of the energies of the several homogeneous 

 masses, and 2Ae s for the sum of the variations of the energies of the 

 several surfaces of discontinuity, the condition of stability may be 



written 



0, (548) 



the total entropy and the total quantities of the several components 

 being constant. The variations to be considered are infinitesimal, 

 but the character A signifies, as elsewhere in this paper, that the 

 expression is to be interpreted without neglect of infinitesimals of the 

 higher orders. Since the temperature and potentials are sensibly 

 constant, the same will be true of the pressures and surface-tensions, 

 and by integration of (86) and (501) we may obtain for any homo- 

 geneous mass 



Ae = t AT; p A v + fa Am x + /z 2 Am 2 + etc., 



and for any surface of discontinuity 



Ae s = t A V 3 + a- As -j- fa Am? + /*f Am 2 + etc. 



These equations will hold true of finite differences, when t, p, &, yu 1? 

 JUL^, etc. are constant, and will therefore hold true of infinitesimal 

 differences, under the same limitations, without neglect of the 



