J. W. Gibhs — Equilibrium of Heterogeneous Substances. 159 



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 *S^j, aS^o, • . . S„ are not all independently variable compo- 

 nents of the given mass, the values which it will he 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 equa- 

 tions connecting J/,, J/g, . . . M„ 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 compo- 

 nents, we have already seen (pages 129-134) that if equilibrium 

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

 properties 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 

 confined to a small part of an indefinitely large fluid mass will cause 

 an ultimate change of state not indefinitely small in degree through- 

 out the whole mass. In the discussion of stability as indicated by 

 fundamental equations it will be convenient to use the term in this 

 sense.* 



* If we wish to know the stability of the given fluid when exposed to a constant tem- 

 perature, or to a constant pressure, or to both, we have only to suppose that there is 

 enclosed in the same envelop with the given fluid another body (which cannot combine 

 with the fluid) of which the fundamental equation is e = Ti], or e = — Pv. or e = Ti] 

 — Pv. as the case may be, (Tand P denoting the constant temperature and pressure, 

 which of course must be those of the given fluid,) and to apply the criteria of page 

 110 to the whole system. When 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 and positive 

 for every other phase of the same components, the value of (133) for the whole system 

 will be less when the system is in its given condition than when it is in any other. 

 (Changes of form and position of the given fluid are of course regarded as immaterial.) 

 Hence the fluid is stable. When it is not possible to assign such values to the con- 

 stants that the value of (133) shall be zero for the given fluid and zero or positive for 

 any other phase, the fluid is of course unstable. In the remaining case, when it is 

 possible to assign such values to the constants that the value of (133) shall be zero 

 for the given fluid and zero or positive for every other phase, but not without the 

 value zero for some other phase, the state of equilibrium of the fluid as stable 

 or neutral wiU be determined by the possibility of satisfying, for any other than 

 the given condition of the fluid, equations like (134), in which, however, the first 

 or the second or both are to be stricken out, according as we are considering the 



