THERMODYNAMICAL SYSTEM OF GIBBS 151 



which /. ^ T, jxi = Ml, H2 = Ms, etc., we may by substituting 

 the value of e given by (205), reduce (204) to the expression 



(P - v)v- (206) 



In order to justify the use of this expression it is necessary to 

 show that in testing the stability of a fluid it is sufficient to take 

 into account only phases for which the temperature and poten- 

 tials are the same as in the given fluid. This can be done by 

 considering the least value of which (201) is capable at a constant 

 value of V. Suppose that (201) has its smallest possible value, 

 without any restriction, when evaluated for a phase having 

 the energy e, entropy 77, volume v, masses Wi, . . .w„.* Then if 

 e', rj', v', m/, rui', . . . m„' are the values referring to any other 

 phase we have 



e' - Tv' + Pv' - Miiui' - M.nii' ... - Af„w„' 

 ^ e — T-q -\- Pv - Mimi — MiiUi ... — Mnirin 



or, if both phases have the same volume, 



€' - e - T(7j' - 77) - Mi{mi' - mi) - Miim^' - roi) . . . ^0. 



Thus if the second phase can be considered as having been 

 formed by an infinitesimal variation of the first phase, at 

 constant volume, we may write this equation as 



de - Tdi) - Midmi - M^dn^ ... ^0. (207) 



But a variation of the energy of the first phase, at constant 

 volume, is given by 



de = tdrj + nidirii + ^l2d'm2 + . . . , (208) 



and (207) and (208) can only both hold if 



t = T, m = Ml, M2 = Mi, etc. 



* It is supposed here that the components of the body are some or all 

 of the components *Si, S2, ■ . -Sn. Gibbs considers the case in which the 

 components of the new phase may be different from those of the given 

 fluid. 



