146 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



requires that the pressure shall be a function of the height alone, 

 and that the density shall be equal to the first derivative of this 

 function, divided by g. 



Conditions (227) contain all that is characteristic of chemical 

 equilibrium. To satisfy these conditions it is necessary and sufficient 



that 



= const. 



fi n -\-gh = const 



.; 



(234) 



The expressions fi v ... jUL n 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 respec,t 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 74-79. 



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 



equation 



vdp= g (m 1 . . . +m n ) dh ; (235) 



or dp=-gydh; (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 dis- 

 continuity. But the condition of equilibrium expressed by equation 



