412 <J. ^V. Gibbs — JEqidlibriwm of Heterogeneous Substances. 



heat to or from external bodies, and in general, to what external 

 influences we are to regard the system as subject. It will be con- 

 venient to suppose that the exterior of the system is fixed, and that 

 neither matter nor heat can be transmitted through it. Other cases 

 may easily be rediic^d to this, or treated in a manner entirely 

 analogous. 



Now if the real system in the given state is unstable, there must be 

 some slightly varied state in which the energy is less, but the entropy 

 and the quantities of the components the same as in the given state, 

 and the exterior of the system unvaried. But it may easily be shown 

 that the given state of the system may be made stable by constrain- 

 ing the surfaces of discontinuity to pass through certain fixed lines 

 situated in the unvaried surfaces. Hence, if the suifaces of discon- 

 tinuity are constrained to jDass through corresponding fixed lines in 

 the surfaces of discontinuity belonging to the varied state just men- 

 tioned, there must be a state of stable equilibrium for the system 

 thus constrained which will differ infinitely little from the given state 

 of the system, the stability of which is in question, and will have the 

 same entropy, quantities of components and exterior, but less energy. 

 The imaginary system will have a similar state, since the real and 

 imaginary systems do not differ in respect to those states which 

 satisfy all the conditions of equilibrium for each surface of discontin- 

 uity. That is, the imaginary system has a state, differing infinitely 

 little from the given state, and with the same entropy, quantities of 

 components, and exterior, but Avith less energy. 



Conversely, if the imaginary system has such a state as that just 

 described, the real system will also have such a state. This may be 

 shown by fixing certain lines in the surfaces of discontinuity of the 

 imaginary system in its state of less energy and then making the 

 energy a minimum under the conditions. The state thus determined 

 will satisfy all the conditions of equilibrium for each surface of dis- 

 continuity, and the real system will therefore have a corresponding 

 state, in which the entropy, quantities of components, and exterior 

 will be the same as in the given state, but the energy less. 



We may therefore determine whether the given system is or is not 

 unstable, by applying the general criterion of instability (V) to the 

 imaginary system. 



If the system is not unstable, the equilibrium is either neutral or 

 stable. Of course we can determine which of these is the case by 

 i-eference to the imaginary system, since tliis determination depends 

 upon states of equilibrium, in regard to Avhich the real and imaginary 



