EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 149 



Condition (244) may be reduced to the form 



ADe-TAD;/-(Y+^ 1 )ADm 1 ...-(Y+^ADm n ^O; (248) 

 and by (246) and (247) to 



ADe - 1 Alty - /*! ADm x . . . - fjL n ADm n ^ 0. (249) 



If values determined subsequently to the change of phase are dis- 

 tinguished by accents, this condition may be written 



De' -tDn'- ^Dm/ . . . - yu n Dm n ' 



-De+tDq + fjL 1 Dm 1 . . . + fJL n Dm n ^ 0, (250) 



which may be reduced by (93) to 



De'-tDri'-[j. l D<m l '...-iUL n Dm n '+pDv^O. (251) 



Now if the element of volume Dv is adjacent to a surface of discon- 

 tinuity, let us suppose De', Drf, Dm/, . . . Dm n ' to be determined (for 

 the same element of volume) by the phase existing on the other side 

 of the surface of discontinuity. As t, fa, . . . ju. n have the same values on 

 both sides of this surface, the condition may be reduced by (93) to 



-p'Dv+pDv^O. (252) 



That is, the pressure must not be greater on one side of a surface of 

 discontinuity than on the other. 



Applied more generally, (251) expresses the condition of equilibrium 

 with respect to the possibility of discontinuous changes of phases at 

 any point. As Dv' = Dv, the condition may also be written 



De' - 1 Dq +p Dv' - j^ Dm/ . . . - fj. n Dm n ' ^ 0, (253) 



which must hold true when t, p, fjL l} . . . fj. n have values determined 

 by any point in the mass, and De', Drf, Dv', Dm/, . . . Dm n ' have values 

 determined by any possible phase of the substances of which the mass 

 is composed. The application of the condition is, however, subject 

 to the limitations considered on pages 74-79. It may easily be shown 

 (see page 104) that for constant values of t, fjL 1} ... fj. n , and of Dv', 

 the first member of (253) will have the least possible value when De', 

 Drf, Dm/, . . . Dm n ' are determined by a phase for which the tempera- 

 ture has the value t, and the potentials the values yUj, ... ju. n . It will 

 be sufficient, therefore, to consider the condition as applied to such 

 phases, in which case it may be reduced by (93) to 



p-p'^0. (254) 



That is, the pressure at any point must be as great as that of any 

 phase of the same components, for which the temperature and the 

 potentials have the same values as at that point. We may also express 

 this condition by saying that the pressure must be as great as is 

 consistent with equations (246), (247). This condition with the 

 equations mentioned will always be sufficient for equilibrium ; when 



