EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 101 



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 considerations 

 similar to those adduced on pages 77-78.) 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 condition in 

 which it has the same entropy and volume. The given condition is 

 therefore stable. (See page 57.) 



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 

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

 page 57.) 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 condition. (A mass consisting of such parts would be in 

 equilibrium, as we have already seen on pages 78, 79.) In this 

 case, if we disregard the quantities connected with the surfaces 

 which divide the homogeneous parts, we must regard the given 

 condition as one of neutral equilibrium. But in regard to these 

 homogeneous parts, which we may evidently consider to be all 

 different phases, the following conditions must be satisfied. (The. 

 accents distinguish the letters referring to the different parts, and 

 the unaccented letters refer to the whole mass.) 



rf+n" + etc. = */, 



v'+t/'+etc. = v, 



m 1 / +m 1 // H-etc. = m 1 , - (134) 



m 2 ' -h m z " + etc. = ra 2 , 

 etc. 



Now the values of rj, v, m 1 , m 2 , etc., are determined by the whole 



fluid mass in its given state, and the values of -, -^>, etc., f, TT 



, v v v v 



etc., f, -, etc., etc., are determined by the phases of the various 



