ABSTRACT BY THE AUTHOR 359 



and thermodynamic state of any such body without regard to its size 

 or form. The word phase has been chosen for this purpose. Such 

 bodies as differ in composition or state are called different phases of 

 the matter considered, all bodies which differ only in size and form 

 being regarded as different examples of the same phase. Phases 

 which can exist together, the dividing surfaces being plane, in an 

 equilibrium which does not depend upon passive resistances to change, 

 are called coexistent. 



The number of independent variations of which a system of co- 

 existent phases is capable is 71+2 r, where r denotes the number of 

 phases, and n the number of independently variable components in 

 the whole system. For the system of phases is completely specified 

 by the temperature, the pressure, and the n potentials, and between 

 these n+2 quantities there are r independent relations (one for each 

 phase), which characterize the system of phases. 



When the number of phases exceeds the number of components by 

 unity, the system is capable of a single variation of phase. The 

 pressure and all the potentials may be regarded as functions of the 

 temperature. The determination of these functions depends upon the 

 elimination of the proper quantities from the fundamental equations 

 in p, t, /z-p yu 2 , etc. for the several members of the system. But 

 without a knowledge of these fundamental equations, the values of 



the differential coefficients such as - may be expressed in terms of 



the entropies and volumes of the different bodies and the quantities 

 of their several components. For this end we have only to eliminate 

 the differentials of the potentials from the different equations of the 

 form (12) relating to the different bodies. In the simplest case, when 

 there is but one component, we obtain the well-known formula 



dp_n'-r[' Q 



dt~v f -v n ~~t(v"-vy 



in which v', v", rf, if' denote the volumes and entropies of a given 

 quantity of the substance in the two phases, and Q the heat which it 

 absorbs in passing from one phase to the other. 



It is easily shown that if the temperature of two coexistent phases 

 of two components is maintained constant, the pressure is in general 

 a maximum or minimum when the composition of the phases is 

 identical. In like manner, if the pressure of the phases is maintained 

 constant, the temperature is in general a maximum or minimum when 

 the composition of the phases is identical. The series of simultaneous 

 values of t and p for which the composition of two coexistent phases 

 is identical separates those simultaneous values of t and p for which 

 no coexistent phases are possible from those for which there are two 

 pairs of coexistent phases. 



