250 



EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



consider those varied surfaces of tension which satisfy the same con- 

 dition. We may therefore regard the surface of tension as determined 

 by v, the volume of one of the homogeneous masses. But the state 

 of the system would evidently be completely determined by the 

 position of the surface of tension and the temperature and potentials, 

 if the entropy and the quantities of the components were variable; 

 and therefore, since the entropy and the quantities of the components 

 are constant, the state of the system must be completely determined 

 by the position of the surface of tension. We may therefore regard 

 all the quantities relating to the system as functions of v', and the 

 condition of stability may be written 



de 7 , , 1 d 2 e 



&**+*- 



where e denotes the total energy of the system. Now the conditions 

 of equilibrium require that 



dv'~ 

 Hence, the general condition of stability is that 



T-75 



dv 2 



(541) 



Now if we write e', e", e s for the energies of the two masses and of 

 the surface, we have by (86) and (501), since the total entropy and 

 the total quantities of the several components are constant, 



de = de' + de" + de 8 = -p'dv' -p"dv" + <rds, 

 or, since dv" = dv', 



de_ 



dv' 



ds 



Hence, 



d 2 e _dp' dp" da- ds 

 dv 7 *' 'M+W^MM 



d 2 s 



(542) 

 (543) 



and the condition of stability may be written 



d 2 s dp' dp" da- ds 

 dv' 2 dv' dv' dv'dv'' 



(544) 



If we now simplify the problem by supposing, as in the similar 

 case on page 245, that we may disregard the variations of the 

 temperature and of all the potentials except one, the condition will 

 reduce to 



70 t T ^ 1 



(545) 



