206 J. W. Glbhs — Equilibrium of Heterogeneous Substances. 



The expressions /^j, . . . /^„ denote quantities which we have called 

 the potentials for the several components, and which are entirely 

 determined at any point in a mass by the nature and state of the 

 mass about that point. We may avoid all confusion between these 

 quantities and the potential of the force of gravity, if we distinguish 

 the former, when necessary, as intrinsic potentials. The relations 

 indicated by equations (234) may then be expressed as follows : 



When a fluid mass is in equilibrium under the influence of gravity^ 

 and has the same independently variable components throughout^ the 

 intrinsic potentials for the several components are constant in any 

 given level, and diminish uniformly as the height increases, the differ- 

 ence of the values of the intrinsic potential for any component at two 

 different levels, being equal to the work done by the force of gravity 

 when a unit of matter falls from the higher to the lower level. 



The conditions expressed by equations (228), (233), (234) are 

 necessary and sufficient for equilibrium, except with respect to the 

 possible formation of masses which are not approximately identical in 

 phase with any previously existing about the points where they may 

 be formed. The possibility of such formations at any point is evidently 

 independent of the action of gravity, and is determined entirely by 

 the phase or phases of the matter about that point. The conditions 

 of equilibrium in this respect have been discussed on pages 128-134. 



But equations (228), (233), and (234) are not entirely independent. 

 For with respect to any mass in which there are no surfaces of dis- 

 continuity (i. e., surfaces where adjacent elements of mass have finite 

 differences of phase), one of these equations will be a consequence of 

 the others. Thus by (228) and (234), we may obtain from (97), 

 which will hold true of any continuous variations of phase, the equa- 

 tion 



V dpz^ — g {m J . . . -f- m„) dh ; (235) 



or dp= - gy dh ; (236) 



which will therefore hold true in any mass in which equations (228) 

 and (234) are satisfied, and in which there are no surfaces of discon- 

 tinuity. But the condition of equilibrium expressed by equation 

 (233) has no exception with respect to surfaces of discontinuity; 

 therefore in any mass in which such surfaces occur, it will be necessary 

 for equilibrium, in addition to the relations expressed by equations 

 (228) and (234), that there shall be no discontinuous change of pressure 

 at these surfaces. 



This superfluity in the particular conditions of equilibrium which 

 we have found, as applied to a mass which is everywhere continuous 



