194 J. W. Gibbs — Equilibrinm of Heterogeneous Substances. 



ON THE VALUES OF THE POTENTIALS WHEN THE QUANTITY OF ONE 

 OF THE COMPONENTS IS VERY SMALL. 



If Ave apply equation (97) to a homogeneous mass having two inde- 

 pendently variable components S^ and S^, and make t, p, and m, 

 constant, we obtain 



i'Ilh\ +mrp] -^0. (210) 



\dm2/t,p, m^ 





or 



Therefore, for ^2=0, either 



f^') =0, (211) 



/^2\ ^ ^_ (212) 



\dm2}t,p, 7/1, 



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 S2, we shall have m2=-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 148, 149) that the 

 value of a diiferential coefficient like that in (212) for any given mass, 

 when the substance S^ (to which ^3 ^"^^ Ma relate) is determined, is 

 independent of the particular substance which we may regard as the 

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

 when the substa.nce denoted by S^ has been so chosen that W2=0, it 

 must hold true without sucli a restriction, Avhich cannot generally 

 be the case. 



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

 ti'ue of any phase which is stalile in regard to continuous changes 

 and in which m^^^O, (/^w^g *'^ capable of negative as icell as positive 

 values. For by (171), in any pliase having that kind of stability, //j 

 is an increasing function of w/ j when t,p, and m.^ are regarded as 

 constant. Hence, //j will have its greatest value when the mass con- 

 sists wholly of aSj, i. e., when mg^rO. Therefore, if w^2 is capable 

 of negative as well as positive values, equation (211) must hold true 

 for rn.^ = 0. (This appears also from the geometrical representation 

 of potentials in the m-t, curve. See page 177.) 



But if Wg 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 physi- 

 cal significance of this case, viz., that an increase of m.^ denotes an 



