EQUILIBKIUM OF HETEROGENEOUS SUBSTANCES. 89 



Other sets of quantities might of course be added which possess 

 the same property. The sets (100), (101), (102) are mentioned on 

 account of the important properties of the quanties \[s, ^, f, and 

 because the equations (88), (90), (92), like (86), afford convenient 

 definitions of the potentials, viz., 



(104) 



r P, * ^dm^t, p , -,n 



etc., where the subscript letters denote the quantities which remain 

 constant in the differentiation, m being written for brevity for all the 

 letters m v m 2 , . . . m n except the one occurring in the denominator. 

 It will be observed that the quantities in (103) are all independent 

 of the quantity of the mass considered, and are those which must, in 

 general, have the same value in contiguous masses in equilibrium. 



On the quantities \[s, x> 



The quantity ^ has been defined for any homogeneous mass by the 



equation 



\l^ tij. (105) 



We may extend this definition to any material system whatever 

 which has a uniform temperature throughout. 



If we compare two states of the system of the same temperature, 

 we have 



V/ - V" = e- e" - t(n' - if). (106) 



If we suppose the system brought from the first to the second of 

 these states without change of temperature and by a reversible 

 process in which W is the work done and Q the heat received by 



the system, then 



e '-e"=TP-Q, (107) 



and W-*')=Q- (108) 



Hence ^/ - \f/ f = W ; (109) 



and for an infinitely small reversible change in the state of the 

 system, in which the temperature remains constant, we may write 



-d\/s = dW. (110) 



Therefore, ^ is the force function of the system for constant 

 temperature, just as e is the force function for constant entropy. 

 That is, if we consider \fr as a function of the temperature and the 

 variables which express the distribution of the matter in space, for 

 every different value of the temperature i/r is the different force 

 function required by the system if maintained at that special 

 temperature. 



