72 BUTLER 



ART. D 



Thus, if there are possible variations which do not satisfy I, 

 there must also be possible variations which do not satisfy II. 

 Thus if condition I is not satisfied, condition II is not satisfied. 

 Conversely, it is shown that if condition II is not satisfied, 

 condition I is not satisfied, so that the two conditions are 

 equivalent to each other. 



4. I nteryr elation of the Conditions* Before proceeding to the 

 proof of the sufficiency and necessity of the criteria of equilib- 

 rium, Gibbs discusses the interpretation of the terms in which 

 the criteria are expressed. 



In the first place, "equations which express the condition of 

 equilibrium, as also its statement in words, are to be inter- 

 preted in accordance with the general usage in respect to differ- 

 ential equations, that is, infinitesimals of higher orders than the 

 first relatively to those which express the amount of change of 

 the system are to be neglected." That is, if be is change in the 

 energy produced by a change bS in the state of the system, and 

 if dt/dS is the limiting value of bt/bS when bS becomes infinitely 

 small, the value of 5e is taken as (de/dS) • bS, infinitesimals of 

 higher orders, such as dh/dS'^, being neglected. Biit different 

 kinds of equilibrium may be distinguished by noting the actual 

 values of the variations. The sign A is used to indicate the 

 value of a variation, when infinitesimals of the higher orders 

 are not neglected. Thus, Ae is the actual energy change pro- 

 duced by a small, but finite variation in the state of the system. 

 The conditions of the different kinds of equilibrium may then 

 be expressed as follows; for stable equilibrium 



(A7?)e < 0, i.e., (Ae), > 0, (6) [3] 



(i.e. the entropy is a maximum at constant energy and the 

 energy a minimum at constant entropy for all possible varia- 

 tions); for neutral equilibrium there must be some variations 

 in the state of the system for which 



(At,), = 0, i.e., (Ae), = 0; (7) [4] 



= Gibbs, I, p. 56, line 38; p. 58, line 40. 



