EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 103 



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

 an ultimate change of state not indefinitely small in degree throughout 

 the whole mass. In the discussion of stability as indicated by funda- 

 mental equations it will be convenient to use the term in this sense.* 



In determining for any given positive values of T and P and any 

 given values whatever of M v M 2 , ... M n whether the expression (133) 

 is capable of a negative value for any phase of the components 

 8 lf S 2 , ... S n , and if not, whether it is capable of the value zero for 

 any other phase than that of which the stability is in question, it 

 is only necessary to consider phases having the temperature T and 

 pressure P. For we may assume that a mass of matter represented 

 by any values of m v ra 2 , . . . m n is capable of at least one state of 

 not unstable equilibrium (which may or may not be a homogeneous 

 state) at this temperature and pressure. It may easily be shown 

 that for such a state the value of e Ttj + Pv must be as small as 

 for any other state of the same matter. The same will therefore be 

 true of the value of (133). Therefore if this expression is capable of 

 a negative value for any mass whatever, it will have a negative value 

 for that mass at the temperature T and pressure P. And if this mass 

 is not homogeneous, the value of (133) must be negative for at least 

 one of its homogeneous parts. So also, if the expression (133) is not 

 capable of a negative value for any phase of the components, any 

 phase for which it has the value zero must have the temperature T 

 and the pressure P. 



*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 = TTJ, or e= Pv, or =Ttj- Pv, 

 as the case may be (T and P denoting the constant temperature and pressure, which 

 of course must be those of the given fluid), and to apply the criteria of page 57 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 will 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 stability 

 of the fluid for constant temperature, or for constant pressure, or for both. The 

 number of coexistent phases will sometimes exceed by one or two the number of the 

 remaining equations, and then the equilibrium of the fluid will be neutral in respect 

 to one or two independent changes. 



