56 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



two different kinds of action upon external systems, the two functions 

 which express the twofold capabilities of the system afford an almost 

 equally simple criterion of equilibrium. 



Criteria of Equilibrium and Stability. 



The criterion of equilibrium for a material system which is isolated 

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

 ing entirely equivalent forms : 



I. For the equilibrium of any isolated system it is necessary and 

 sufficient that in all possible variations of the 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 r\ 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 



05^0. (1) 



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

 sufficient that in all 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 



... '' <&),SO. (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 (1) is not satisfied, there must be some variation in the 

 state of the system for which 



<ty>0 and (5e = 0; 



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) 



<fy = and <te<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 



<Je<0 and cty = 0; 



hence tfcere must also be one for which 



<$e = and &/>0; 



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 



