EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 173 



only a part of the phases are actually capable of existing, we might 

 still suppose the particular phases which alone can exist to be deter- 

 mined by some other principle than that of the free convertibility 

 of the components (as if, perhaps, the case were analogous to one of 

 constraint in mechanics), it may easily be shown that such a hypothesis 

 is entirely untenable, when the quantities of the proximate components 

 may be varied independently by suitable variations of the temperature 

 and pressure, and of the quantities of the ultimate components, and 

 it is admitted that the relations between the energy, entropy, volume, 

 temperature, pressure, and the quantities of the several proximate 

 components in the gas-mixture are the same as for an ordinary ideal 

 gas-mixture, in which the components are not convertible. Let us 

 denote the quantities of the ri proximate components of a gas-mixture 

 A by m^ m z , etc., and the quantities of its n ultimate components by 

 m lt m 2 , etc. (n denoting a number less than n'), and let us suppose 

 that for this gas-mixture the quantities e, ?/, v, t, p, m 1? m 2 , etc. satisfy 

 the relations characteristic of an ideal gas-mixture, while the phase of 

 the gas-mixture is entirely determined by the values of m 1} n^, etc., 

 with two of the quantities e, 77, v, t, p. We may evidently imagine 

 such an ideal gas-mixture B having n' components (not convertible), 

 that every phase of A shall correspond with one of B in the values of 

 e, q, v, t, p, m x , m 2 , etc. Now let us give to the quantities m 1 , m 2 , etc. 

 in the gas-mixture A any fixed values, and for the body thus defined 

 let us imagine the v-q-e surface (see page 116) constructed; likewise 

 for the ideal gas-mixture B let us imagine the v-q-e surface constructed 

 for every set of values of m 1? ra 2 , etc. which is consistent with the 

 given values of m^ m 2 , etc., i.e., for every body of which the ultimate 

 composition would be expressed by the given values of m 1 ,m 2 , etc. It 

 follows immediately from our supposition, that every point in the 

 v-jj-6 surface relating to A must coincide with some point of one of 

 the v-rj-e surfaces relating to B not only in respect to position but also 

 in respect to its tangent plane (which represents temperature and 

 pressure) ; therefore the v-r\-e surface relating to A must be tangent to 

 the various v-q- surfaces relating to B, and therefore must be an 

 envelop of these surfaces. From this it follows that the points which 

 represent phases common to both gas-mixtures must represent the 

 phases of dissipated energy of the gas-mixture B. 



The properties of an ideal gas-mixture which are assumed in regard 

 to the gas-mixture of convertible components in the above demonstra- 

 tion are expressed by equations (277) and (278) with the equation 



e = I tl (c 1 m 1 t+m l E 1 ). (324) 



It is usual to assume in regard to gas-mixtures having convertible 

 components that the convertibility of the components does not affect 



