EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 97 



Or, when the r bodies considered have not the same independently 

 variable components, if we still denote by n the number of inde- 

 pendently variable components of the r bodies taken as a whole, the 

 number of independent variations of phase of which the system is 

 capable will still be n + 2 r. In this case, it will be necessary to 

 consider the potentials for more than n component substances. Let 

 the number of these potentials be n + h. We shall have by (97), as 

 before, r relations between the variations of the temperature, of the 

 pressure, and of these n+h potentials, and we shall also have by (43) 

 and (51) h relations between these potentials, of the same form as the 

 relations which subsist between the units of the different component 

 substances. 



Hence, if r = n+2, no variation in the phases (remaining coex- 

 istent) is possible. It does not seem probable that r can ever exceed 

 n + 2. An example of n = 1 and r = 3 is seen in the coexistent solid, 

 liquid, and gaseous forms of any substance of invariable composition. 

 It seems not improbable that in the case of sulphur and some other 

 simple substances there is more than one triad of coexistent phases; 

 but it is entirely improbable that there are four coexistent phases of 

 any simple substance. An example of n = 2 and r = 4 is seen in a 

 solution of a salt in water in contact with vapor of water and two 

 different kinds of crystals of the salt. 



Concerning n + l Coexistent Phases. 



We will now seek the differential equation which expresses the 

 relation between the variations of the temperature and the pressure in 

 a system of n + 1 coexistent phases (n denoting, as before, the number 

 of independently variable components in the system taken as a whole). 



In this case we have n + l equations of the general form of (97) 

 (one for each of the coexistent phases), in which we may distinguish 

 the quantities q, v, ra p m 2 , etc., relating to the different phases by 

 accents. But t and p will each have the same value throughout, and 

 the same is true of yu 1? jn 2 , etc., so far as each of these occurs in the 

 different equations. If the total number of these potentials is n+h, 

 there will be h independent relations between them, corresponding to 

 the h independent relations between the units of the component 

 substances to which the potentials relate, by means of which we 

 may eliminate the variations of h of the potentials from the equations 

 of the form of (97) in which they occur. 



Let one of these equations be 



v'dp = r)'dt + m a 'djUL a +m b 'd[jL b + etc., (124) 



and by the proposed elimination let it become 



dp n > (125) 



G.I. G 



