148 BUTLER ART. D 



it is evident that the fluid is unstable. It may also happen 

 that while T, P, Mi, Af 2, • • • Mn niay be given such values that 

 (201) is zero for the given fluid there is some other phase for 

 which (201) is also zero. This other phase must obviously 

 have the same temperature and pressure, and the same values of 

 the potentials, and is therefore a phase which could coexist with 

 the given fluid. But Gibbs points out that although there 

 may be phases which can coexist with the given mass, it is 

 highly improbable that such phases could be formed within 

 the given mass without a change of entropy or of volume. 

 Thus although at the triple point water can coexist with ice 

 and vapor, a quantity of water in this state enclosed in an 

 envelop which has a constant volume and is impervious to heat 

 is quite stable. 



27. Condition of Stability at Constant Temperature and 

 Pressure. In considering whether (201) is capable of a negative 

 value for any phase, Gibbs points out that it is only necessary 

 to consider phases which have the temperature T and the 

 pressure P. For it may be assumed that the mass is capable 

 of at least one state of not unstable equilibrium at this tem- 

 perature and pressure, and in such a state the value of (201) 

 must be as small as for any other state of the same matter. 

 Therefore, if (201) is capable of a negative value, it wUl have a 

 negative value at the temperature T and the pressure P. Also, 

 if it is not capable of a negative value, any state for which it 

 has the value zero must have the temperature T and the pressure 

 P. 



For any body at the temperature T and the pressure P, (201) 

 reduces to 



r - MiMi - Minh ... - M„w„, (202) [135] 



and in this form is capable of a very direct application, which is 

 the basis of the geometrical methods employed by Gibbs in his 

 use of curves and surfaces. 



Consider a series of homogeneous phases containing the two 

 components Si and *S2 in different proportions. The ^-curve for 

 a constant temperature t and pressure p is obtained by plotting 



