78 BUTLER 



ART. D 



which we have reason to expect. Such considerations seem to 

 justify us in regarding such a state as we are discussing as one 

 of theoretical equihbrium; although as the equilibrium is evi- 

 dently unstable, it cannot be realized." 



The argument of the last section is here applied to the higher 

 differential coefficients of the quantities which represent the 

 state of the system with respect to the time. Thus if <S is one 

 of the quantities representing the state of the system, it is shown 

 that all such differential coefficients as 



dt 



etc.. 



are zero in the state for which (5r?)« ^ 0. 



It is evident that the system cannot be in equilibrium unless 

 all these quantities have the value 0, for if dS/dt is zero in the 

 initial state and one of the higher coefficients has a finite value, 

 dS/dt will have a finite value at a subsequent time. The proof 

 that they are zero in the state for which (Stj)^ ^ may be stated 

 in greater detail as follows. If any of the differential coefficients 

 have finite values, the system must undergo a change, which, 

 however, may be infinitely slow so long as (677) « ^ 0. But, by an 

 infinitesimal modification in the circumstances, it will be pos- 

 sible to produce a state for which (8T])t < 0. Such changes 

 will then be impossible. That is, an infinitely small modifica- 

 tion of the circumstances will cause a finite change in the 

 values of those differential coefficients which previously had 

 finite values. But this is regarded as impossible. The sys- 

 tem can therefore continue unchanged in the state for which 

 (8r))t ^ 0, which must be regarded as a state of equihbrium, 

 but since there are changes for which (Atj)^ > 0, it is evidently a 

 state of unstable equilibrium. 



6. Necessity of the Criteria of Equilihrium* When "the active 

 tendencies of the system are so balanced that changes of every 

 kind, except those excluded in the statement of the condition of 

 equilibrium, can take place reversibly (i.e., both in the positive 

 and the negative direction,) in states of the system differing 



* Gibhs, I, p. 61, line 11 ; p. 62, line 8. 



