J. W. Gihbs—Equilibriu7n of Heterogeneous Substances. 157 



Now, if it is possible to assign such values to the constants T, F, 

 M^, J/2, . . . 3f„ that the value of the expression 



^ - T,/-\-Pv ~ J/, m, - 3/2 W2 . . . - 3f„m„ (133) 



shall be zero for the given fluid, and shall be positive for every other 

 phase of the same ' components, i. e., for every homogeneous body* 

 not identical in nature and state with the given fluid (but composed 

 entirely oi S^, S^, . . . /S„), the condition of the given fluid will be 

 stable. 



For, in any condition whatever of the given mass, whether or not 

 homogeneous, or fluid, if the value of the expression (133) is not 

 negative for any homogeneous part of the mass, its value for the 

 whole mass cannot be negative ; and if its value cannot be zero for 

 any homogeneous part which is not identical in phase with the mass 

 in its given condition, its value cannot be zero for the whole except 

 when the whole is in the given condition. Therefore, in the case 

 supposed, the value of this expression for any other than the given 

 condition of the mass is positive. (That this conclusion cannot be 

 invalidated by the fact that it is not entirely correct to regard a 

 composite mass as made up of homogeneous parts having the 

 same properties in respect to energy, entropy, etc., as if they were 

 parts of larger homogeneous masses, will easily appear from consider- 

 ations similar to those adduced on pages 131-133.) If, then, the 

 value of the expression (133) for the mass considered is less when it 

 is in the given condition than when it is in any other, the energy of 

 the mass in its given condition must be less than in any other condi- 

 tion in which it has the same entropy and volume. The given con- 

 dition is therefore stable. (See page 110.) 



Again, if it is possible to assign such values to the constants in 

 (133) that the value of the expression shall be zero for the given 

 fluid mass, and shall not be negative for any phase of the same com- 

 ponents, the given condition will be evidently not unstable. (See 

 page 110.) It will be stable unless it is possible for the given matter 

 in the given volume and with the given entropy to consist of homo- 

 geneous parts for all of which the value of the expression (133) is zero, 

 but which are not all identical in phase with the mass in its given con- 

 dition. (A mass consisting of such parts would be in equilibrium, as 

 we have already seen on pages 133, 134.) In this case, if we disre- 

 gard the quantities connected with the surfaces which divide the 



* A vacuum is throughout this discussion to be regarded as a limiting case of an 

 extremely rarified body. We may thus avoid the necessity of the specific mention of a 

 vacimm in propositions of this kind. 



