J. W. Gihhs — Equilibrium of Heterogeneous Substances. 109 



CRITERIA OP EQUlLIBRIUiM AND STABILITY. 



Tlie criterion of equilibrium for a material system Avhicli is isolated 

 from all external influences may be expressed in either of the follow- 

 ing entirely equivalent forms : 



I. M>r the equilibrium of any isolated si/stem it is necesmn/ and 

 sufficient that in all possible variations of tlie state of the system 

 which do not alter its energy^ the variation of its entropy shall either 

 vanish or be negative. If e denote the energy, and ;/ the entropy of 

 the system, and we use a subscript letter after a variation to indicate 

 a quantity of which the value is not to be varied, the condition of 

 equilibrium may be written 



{^V)e ^0- (1) 



II. For the equilibrium of any isolated system it is 7iecessary and 

 sufficient that in cdl possible variations in the state of the system 

 which do not alter its entropy^ the variation of its energy shall either 

 vanish or be positive. This condition may be written 



(d>),^ 0. (2) 



That these two theorems are equivalent will appear from the con- 

 sideration that it is always possible to increase both the energy and 

 the entropy of the system, or to decrease both together, viz., by 

 imparting heat to any part of the system or by taking it away. For, 

 if condition (l) is not satisfied, there must be some variation in the 

 state of the system for which 



6i] > and de =zQ; 

 therefore, by diminishing both the energy and the entropy of the 

 system in its varied state, we shall obtain a state for which (considered 

 as a variation from the original state) 



6i]z=i and (^f <0; 

 therefore condition (2) is not satisfied. Conversely, if condition (2) 

 is not satisfied, there must be a variation in the state of the system 



for which 



(Jf < and 6i]^^0\ 



hence there must also be one for which 



^f rz: and (J// > ; 



therefore condition (1) is not satisfied. 



The equations which express the condition of equilibrium, as also 

 its statement in words, are to be interpreted in accordance with the 

 o-eneral usage in respect to differential equations, that is, infinitesimals 



