EQUILIBKIUM OF HETEROGENEOUS SUBSTANCES. 249 



will satisfy all the conditions of equilibrium for each surface of 

 discontinuity, 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 (7) 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 

 reference to the imaginary system, since the determination depends 

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

 systems do not differ. We may therefore determine whether the 

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

 applying the criteria (3)-(7) to the imaginary system. 



The result which we have obtained may be 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 surfaces, even 

 when the condition of equilibrium relating to pressure {equation (500)} 

 is not satisfied. 



It follows immediately from this principle, in connection with 

 equations (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 con- 

 ditions of stable or neutral equilibrium, when these are satisfied, it 

 is only necessary 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 described by apply- 

 ing it to a problem but slightly different from one already (pp. 244, 

 245) 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 



