264 Dr. S. A. Shorter on the Constitution 



of length (dv/d<r) 9r 8o; a distance (dp/dr) Q $T apart. Now the 



perimeter must be described in an an ti- clockwise direction. 

 Hence the above two differential coefficients must have the 

 same sign ; so that equating the areas of the parallelograms 

 PQRS and P'Q'B/S', we obtain the relation 



\da) 9r ,=Z \dp) 9 (1) 



Since in a binary system the constancy of* r and 6 is 

 equivalent to the constancy of p and #, this equation may 

 be written * 



\da) pQ ~\dpjQ ) 



We see from this equation that if the surface-tension 

 depends upon the pressure, it is necessary to change the 

 volume, when the area of the surface is changed, in order 

 to keep unchanged the nature of the different parts of the 

 system. Such a change would obviously be unnecessary if 

 the two phases were homogeneous right up to the surface 

 of separation. Let p and p 1 denote the densities in the 

 (p phase, and p ' and p^ those in the <f> ! phase, of C and C\ 

 respectively. Let V and V denote the respective volumes 

 of the phases, and M and M 7 the respective total masses of 

 the components. We see that in general the equations 



M = poV+po'Y', 

 M, = Pl Y+ Pl 'T, 



will not be verified. Let us write 



<rr o = M -( /3o V+p 'V), .... (3) 

 ar^M.-^V + ^'V) (4) 



We will call (following Milner |) each of the quantities 

 r and Fj (defined by the above equations) the " surface 

 excess " of the corresponding component. It is evident 

 that r and I\ will depend only on the nature of the system, 

 and not on the dimensions of its parts. 



The differential coefficient {dv/d(r) e is easily evaluated in 



* This relation may be deduced more concisely, though in a less 

 elementary manner, from the equation dJJ=9dS— pdv-\-rdar (U = energv, 

 S = entropy), by writing it in the form d(U - 9S-\-pv) = — Sd9-\-vdp-\-rda: 



t Loc. cit. Gibbs uses the term " surface density." 



