248 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



attribute in its initial state the same homogeneous masses and surfaces 

 of discontinuity which belong to the real system. We shall also 

 suppose that the homogeneous masses and surfaces of discontinuity of 

 this system, which we may call the imaginary system, have the same 

 fundamental equations as those of the real system. But the imaginary 

 system is to differ from the real in that the variations of its state are 

 limited to such as do not violate the conditions of equilibrium relating 

 to temperature and the potentials, and that the fundamental equations 

 of the surfaces of discontinuity hold true for these varied states, 

 although the condition of equilibrium expressed by equation (500) 

 may not be satisfied. 



Before proceeding farther, we must decide whether we are to 

 examine the question of stability under the condition of a constant 

 external temperature, or under the condition of no transmission of 

 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 reduced 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 constraining 

 the surfaces of discontinuity to pass through certain fixed lines situated 

 in the unvaried surfaces. Hence, if the surfaces of discontinuity are 

 constrained to pass through corresponding fixed lines in the surfaces 

 of discontinuity belonging to the varied state just mentioned, 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 discontinuity. 

 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 with 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 



