J. W. Gibbs — JSquilibHuni of Heteroffe)ieous Substances. 209 



Condition (244) may be reduced to the form 

 ADe^ TJDj] - {r-\-M^)JBm, ..._(]"+ ]\QJJ)m„^0; (248) 

 and by (246) and (247) to 



JDe - tJDj) - //, JBm^ ... — //„ JZ>^/?„^ 0. (249) 



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

 guished by accents, this condition may be written 

 J)s' - t Df/ - //j Diu^' ... - /.i„Brn„' 



— Be + t D)i -{- 1.1^ Bm^ ... + //„ Bm^ 0, (250) 

 which may be reduced by (93) to 



Be' - tB)]' - //, Bm^, ... - j.i„Bi>i„' -]- pBv^O. (251) 



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

 tinuity, let us suppose Bi\ Bif, Bm^\ . . . Bm„' to be determined 

 (for the same element of volume) by the phase existing on the other 

 side of the surface of discontinuity. As ^, //,,.. . //„ have the same 

 values on both sides of this surface, the condition may be reduced by 

 (93) to 



— p'Bv +pBv^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 Bv' = Bv, the condition may also be written 



Be' - tB}/ +pBij' - yt<, i>m,', ... - u„Bm„'^0, (253) 

 which must hold true when t, p, /a^, . . . //„ have values determined 

 by any point in the mass, and Ba', Bt/, Bv', Btn^ , . . . BmJ, 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 128-134. It may 

 easily be shown (see pages 160, 161) that for constant values of t, //,, 

 . . . //„, and of Bv' , the first member of (253) will have the least possi- 

 ble value when Be', Bif, Bm j ', . . . Bm^ are determined by a phase 

 for which the temperature has the value t, and the potentials the 

 values yt<,, . . . //„. 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'^O. (254) 



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



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



Trans. Conn. Acad., Vol. III. 27 April, 1876. 



