EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 135 



On the Values of the Potentials when the Quantity of one of 

 the Components is very small. 



If we apply equation (97) to a homogeneous mass having two 

 independently variable components S 1 and S z , and make t, p, and m l 

 constant, we obtain 



=0. (210) 



P> , Bl fly i, p, mi 



Therefore, for ra 2 = 0, either 



= 0, (211) 



or Kl =00 . (212) 



Pi 



Now, whatever may be the composition of the mass considered, we 

 may always so choose the substance /S^ that the mass shall consist 

 solely of that substance, and in respect to any other variable com- 

 ponent S 2 , we shall have ra 2 = 0. But equation (212) cannot hold true 

 in general as thus applied. For it may easily be shown (as has been 

 done with regard to the potential on pages 92, 93) that the value of 

 a differential coefficient like that in (212) for any given mass, when 

 the substance S 2 (to which ra 2 and // 2 relate) is determined, is inde- 

 pendent of the particular substance which we may regard as the other 

 component of the mass ; so that, if equation (212) holds true when the 

 substance denoted by S l has been so chosen that m 2 = 0, it must hold 

 true without such a restriction, which cannot generally be the case. 



In fact, it is easy to prove directly that equation (211) will hold 

 true of any phase which is stable in regard to continuous changes and 

 in which m 2 = 0, if m 2 is capable of negative as well as positive values. 

 For by (171), in any phase having that kind of stability, fa is an 

 increasing function of ra x when t, p, and m 2 are regarded as constant. 

 Hence, /z x will have its greatest value when the mass consists wholly 

 of S lt i.e., when m 2 = 0. Therefore, if m 2 is capable of negative as well 

 as positive values, equation (211) must hold true for m 2 = 0. (This 

 appears also from the geometrical representation of potentials in the 

 -m-f curve. See page 119.) 



But if m 2 is capable only of positive values, we can only conclude 

 from the preceding considerations that the value of the differential 

 coefficient in (211) cannot be positive. Nor, if we consider the 

 physical significance of this case, viz., that an increase of m 2 denotes 

 an addition to the mass in question of a substance not before 

 contained in it, does any reason appear for supposing that this 

 differential coefficient has generally the value zero. To fix our 



