J. W. Gibhs — Equilibrium of Heterogeneous Substances. 399 



The same physical relations may of course be deduced without 

 giving up the use of the surface of tension as a dividing surface, but 

 the formulae which express them will be less simple. If we make 

 ^ ;<3, /'4, etc. constant, Av^e have by (98) and (508) 



dp" =y," d/.i^ -f y/ dMs, 

 da = — I\ d/.{^ — 1\ djji^, 



where we may suppose F^ and /"g to be determined with reference 

 to the surface of tension. Then, if dp' =.dp)" ., 



and 



That is, 



dff = /', — -; ~„ du^ — r^ diu. 



Yx -r, . 





The reader will observe that — r— ' — -r. represents the distance be- 



Yx -Yx 

 tween the surface of tension and that dividing surface which would 



make 7^^ = ; the second number of the last equation is therefore 



equivalent to — T^g-d)- 



If any component substance has the same density in the two homo- 

 geneous masses separated by a plane surfixce of discontinuity, the 

 value of the superficial density for that component is independent 

 of the position of the dividing surface. In this case alone we may 

 derive the value of the superficial density of a component with 

 reference to the surface of tension from the fundamental equation for 

 plane surfaces alone. Thus in tlie last equation, when y^ =■ y ^ •, tbe 

 second member will reduce to — T^. It will be observed that to 

 make jt)'— -^^", ^, /Y3, /i^, etc. constant is in this case equivalent to 

 making ^, /^,, //g, z^^, etc. constant. 



Substantially the same is true of the superficial density of entropy 

 or of energy, when either of these has the same density in the two 

 homogeneous masses.* 



* "With respect to questions which concern only the form of surfaces of discontinuity, 

 such precision as we have employed in regard to the position of the dividing surface 

 is evidently quite unnecessary. This precision has not been used for the sake of the 

 mechanical part of the problem, which does not require the surface to be defined 

 with greater nicety than we can employ in our observations, but in order to give 



Trans. Conn. Acad., Vol. III. 51 July, 1877. 



