102 EQUILIBKIUM OF HETEROGENEOUS SUBSTANCES. 



parts. But the phases of these parts are evidently determined by 

 the phase of the fluid as given. They form, in fact, the whole set of 

 coexistent phases of which the latter is one. Hence, we may regard 

 (134) as n -f 2 linear equations between v' t v", etc. (The values of 

 v', v", etc., are also subject to the condition that none of them can be 

 negative.) Now one solution of these equations must give us the 

 given condition of the fluid; and it is not to be expected that they 

 will be capable of any other solution, unless the number of different 

 homogeneous parts, that is, the number of different coexistent phases, 

 is greater than 7i + 2. We have already seen (page 97) that it is 

 not probable that this is ever the case. 



We may, however, remark that in a certain sense an infinitely large 

 fluid mass will be in neutral equilibrium in regard to the formation 

 of the substances, if such there are, other than the given fluid, for 

 which the value of (133) is zero (when the constants are so deter- 

 mined that the value of the expression is zero for the given fluid, 

 and not negative for any substance); for the tendency of such a 

 formation to be reabsorbed will diminish indefinitely as the mass 

 out of which it is formed increases. 



When the substances 8 lf S 2 , . . . S n are all independently variable 

 components of the given mass, it is evident from (86) that the con- 

 ditions that the value of (133) shall be zero for the mass as given, 

 and shall not be negative for any phase of the same components, 

 can only be fulfilled when the constants T, P, M v M 2 , . . . M n are equal 

 to the temperature, the pressure, and the several potentials in the 

 given mass. If we give these values to the constants, the expression 

 (133) will necessarily have the value zero for the given mass, and we 

 shall only have to inquire whether its value is positive for all other 

 phases. But when 8 V S 2 , ... S n are not all independently variable 

 components of the given mass, the values which it will be necessary 

 to give to the constants in (133) cannot be determined entirely from 

 the properties of the given mass ; but T and P must be equal to its 

 temperature and pressure, and it will be easy to obtain as many 

 equations connecting M v M 2 , . . . M n with the potentials in the given 

 mass as it contains independently variable components. 



When it is not possible to assign such values to the constants in 

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

 and either zero or positive for any phase of the same components, 

 we have already seen (pages 75-79) that if equilibrium subsists 

 without passive resistances to change, it must be in virtue of pro- 

 perties which are peculiar to small masses surrounded by masses 

 of different nature, and which are not indicated by fundamental 

 equations. In this case, the fluid will necessarily be unstable, if we 

 extend this term to embrace all cases in which an initial disturbance 



