J. W. Gibbs — Equilibrium of Heterogeneous Substances. 413 



systems do not differ. We niny therefore determine whether the 

 equilibrium of the given system is stable, neutral, or unstable, by 

 applying the criteria (3)-(V) to the imaginary system. 



The result which we have obtained maybe expressed as follows:^ 

 In applying to a fluid system which is in equilibrium, and of which 

 all the small parts taken separately are stable, the criteria of stable, 

 neutral, and unstable equilibrium, we may regard the system as 

 under constraint to satisfy the conditions of equilibrium relating to 

 temperature and the potentials, and as satisfying the relations ex- 

 pressed by the fundamental equations for masses and siirfaces, even 

 when the condition of equilibrium relating to pressure [equation 

 (500)] is not satisfied. 



It follows immediately from this principle, in connection with equa- 

 tions (501) and (86), that in a stable system each surface of tension 

 must be a surface of minimum area for constant values of the volumes 

 which it divides, when the other surfaces bounding these volumes 

 and the perimeter of the surface of tension are regarded as fixed ; 

 that in a system in neutral equilibrium each surface of tension will 

 have as small an area as it can receive by any slight variations under 

 the same limitations ; and that in seeking the remaining conditions of 

 stable or neutral equilibrium, when these are satisfied, it is only 

 necessaiy to consider such varied surfaces of tension as have similar 

 properties with reference to the varied volumes and perimeters. 



We may illustrate the method which has been desci-ibed by apply- 

 ing it to a problem but slightly different from one already (pp. 408, 

 409) discussed by a different method. It is required to determine the 

 conditions of stability for a system in equilibrium, consisting of two 

 different homogeneous masses meeting at a surface of discontinuity, 

 the perimeter of which is invariable, as well as the exterior of the 

 whole system, which is also impermeable to heat. 



To determine what is necessary for stability in addition to the 

 condition of minimum area for the surface of tension, we need only 

 consider those varied surfaces of tension w^hich satisfy the same con- 

 dition. We may therefore regard the surface of tension as deter- 

 mined by w', the volume of one of the homogeneous masses. But the 

 state of tlie system would evidently be completely determined by the 

 position of the surface of tension and the temperature and potentials, 

 if the entropy and the quantities of the components were variable; 

 and therefore, since the entropy and the quantities of the components 

 are constant, the state of the system must be completely determined 

 by the position of the surface of tension. We may therefore regard 



