J. W. Gihbs — EquiUhrlum of Heterogeneous ISuhstances. 149 



The value of the two expressions is, however, the same, for, although 

 ?>?„• is not equal to w„,, we may of course suppose chuyy to he equal to 

 clm^, and then the numerators in the two fractions will also be equal, 

 as they each denote the increase of energy of the liquid, when the 

 quantity (hn^y or drn^, of water is added without altering the entropy 

 and volume of the liquid. Precisely the same considerations will 

 apply to any other case. 



In fact, we may give a definition of a potential which shall not pre- 

 suppose any choice of a particular set of substances as the components 

 of the homogeneous mass considered. 



Definition. — If to any homogeneous mass we suppose an infinitesi- 

 mal quantity of any substance to be added, the mass remaining 

 homogeneous and its entropy and volume remaining unchanged, the 

 increase of the energy of the mass divided by the quantity of the 

 substance added is the potential for that substance in the mass con- 

 sidered. (For the purposes of this definition, any chemical element or 

 combination of elements in given proportions may be considered a 

 substance, whether capable or not of existing by itself as a homoge- 

 neous body.) 



In the above definition we may evidently substitute for entropy, 

 volume, and energy, respectively, either temperature, volume, and 

 the function ij- ; or entropy, pressure, and the function x ; or tempera- 

 ture, pressure, and the function ;;. (Compare equation (104).) 



In the same homogeneous mass, therefore, we may distinguish the 

 potentials for an indefinite number of substances, each of which has a 

 perfectly determined value. 



Between the potentials for diiferent substances in the same homo- 

 geneous mass the same equations will subsist as between the units 

 of these siibstances. That is, if the substances, *S„, /S',„ etc., ^S^, Si, etc., 

 are components of any given homogeneous mass, and are such that 



a 2„ + /^ g, + etc. = n ©, + ^ ©/+ etc., (120) 



©a, S45 etc., 2i, S/, etc. denoting the units of the several substances, 

 and «, /j, etc., «, A, etc. denoting numbers, then if /<„, ^,„ etc., /z^, /^„ 

 etc. denote the potentials for these substances in the homogeneous 



mass, 



a i-ia + /^ yWi + etc. = H f-ik + A /^, + etc. (121) 



To show this, we will suppose the mass considered to be very large. 

 Then, the first number of (121) denotes the increase of the energy of 

 the mass produced by the addition of the matter represented by the 

 first member of (120), and the second member of (121) denotes the 



