EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 229 



when the bounding surface is fixed, and the total entropy and total 

 quantities of the various components are constant. We may suppose 



V s , n'> n"> m ?> m i' m i"> m f> m 2'> m 2"> e ^ c -' t k 6 ft U constant. Then 

 by (497) and (12) the condition reduces to 



a- 8s -p'Sv' -p"Sv" = 0. (499) 



(We may set = for ^, since changes in the position of the dividing 

 surface can evidently take place in either of two opposite directions.) 

 This equation has evidently the same form as if a membrane without 

 rigidity and having a tension or, uniform in all directions, existed 

 at the dividing surface. Hence the particular position which we 

 have chosen for this surface may be called the surface of tension, and 

 <r the superficial tension. If all parts of the dividing surface move a 

 uniform normal distance SN, we shall have 



to = (<?! + c 2 )s SN, Sv' = s SN, Sv" =-sSN; 

 whence <r(c l +c z )=p' p", (500) 



the curvatures being positive when their centers lie on the side to 

 which p' relates. This is the condition which takes the place of that 

 of equality of pressure (see pp. 65, 74) for heterogeneous fluid 

 masses in contact, when we take account of the influence of the 

 surfaces of discontinuity. We have already seen that the conditions 

 relating to temperature and the potentials are not affected by these 

 surfaces. 



Fundamental Equations for Surfaces of Discontinuity between 



Fluid Masses. 



In equation (497) the initial state of the system is supposed to be 

 one of equilibrium. The only limitation with respect to the varied 

 state is that the variation shall be reversible, i.e., that an opposite 

 variation shall be possible. Let us now confine our attention to 

 variations in which the system remains in equilibrium. To dis- 

 tinguish this case, we may use the character d instead of S, and write 



de 8 = t drj B + a-ds+[jL l dmf + JUL Z dm% -f etc. (501 ) 



Both the states considered being states of equilibrium, the limitation 

 with respect to the reversibility of the variations may be neglected, 

 since the variations will always be reversible in at least one of the 

 states considered. 



If we integrate this equation, supposing the area s to increase from 

 zero to any finite value s, while the material system to a part of 

 which the equation relates remains without change, we obtain 



(502) 



