78 EQUILIBEIUM OF HETEROGENEOUS SUBSTANCES. 



the condition is not satisfied. It is evident that the value of the 

 expression e -Tr,+Pv-M l m l -M^m,_...-M n m n (57) 



applied to a mass like including some very small masses like N, 

 will be negative, and will decrease if the number of these masses like 

 N is increased, until there remains within the whole mass no portion 

 of any sensible size without these masses like N, which, it will be 

 remembered, have no sensible size. But it cannot decrease without 

 limit, as the value of (54) cannot become infinite. Now we need not 

 inquire whether the least value of (57) (for constant values of T, P y 

 M v M 2 , . . . M n ) would be obtained by excluding entirely the mass 

 like 0, and filling the whole space considered with masses like N, 

 or whether a certain mixture would give a smaller value, it is 

 certain that the least possible value of (57) per unit of volume, and 

 that a negative value, will be realized by a mass having a certain 

 homogeneity. If the new part N for which the condition (52) is not 

 satisfied occurs between two different original parts 0' and 0", the 

 argument need not be essentially varied. We may consider the 

 value of (57) for a body consisting of masses like 0' and 0" separated 

 by a lamina N. This value may be decreased by increasing the 

 extent of this lamina, which may be done within a given volume 

 by giving it a convoluted form ; and it will be evident, as before, 

 that the least possible value of (57) will be for a homogeneous mass, 

 and that the value will be negative. And such a mass will be not 

 merely an ideal combination, but a body capable of existing, for as the 

 expression (57) has for this mass in the state considered its least 

 possible value per unit of volume, the energy of the mass included in 

 a unit of volume is the least possible for the same matter with the 

 same entropy and volume, hence, if confined in a non-conducting 

 vessel, it will be in a state of not unstable equilibrium. Therefore 

 when (50), (51), and (43) are satisfied, if the condition (52) is not 

 satisfied in regard to all possible new parts, there will be some homo- 

 geneous body which can be formed out of the substances 8 V S 2 , ... S n 

 which will not satisfy condition (53). 



Therefore, if the initially existing masses satisfy the conditions (50), 

 (51), and (43), and condition (53) is satisfied by every homogeneous 

 body which can be formed out of the given matter, there will be 

 equilibrium. 



On the other hand, (53) is not a necessary condition of equilibrium. 

 For we may easily conceive that the condition (52) shall hold true 

 (for any very small formations within or between any of the given 

 masses), while the condition (53) is not satisfied (for all large masses 

 formed of the given matter), and experience shows that this is very 

 often the case. Supersaturated solutions, superheated water, etc., 



