418 >/. W. Gibbs — Equilibrkmi of Heterogeneous Substances. 



excess of the total energy, entropy, etc. in and about the globular 

 mass above what would be in the same space if it were uniformly 

 filled with matter of the phase of the exterior mass, we shall have 

 necessai-ily with reference to the whole dividing surface 



m\ = [m^] - v' (;/,' — ;/,"), m| = [m^] - v' {y ^ — y ^\ etc., 



where fy', ^v", 'a', 'a", k/, y y •, etc. denote, in accordance with our 

 usage elsewhere, the volume-densities of energy, of entropy, and of 

 the various components, in the two homogeneous masses. We may 

 thus obtain from equation (502) 



0- 5 = [f] - v' (fv' - fv") - t L'/] + i ^' ('/v' - Vv") 



-l.i^[_m^'\+fi^v'{y^'-y^")-H2\:m2\+f-i2v'{y^'-y./)-iilc. (551) 

 But by (93), 



p' — — f v' + 1 ?h' + y" 1 r I ' + /^2 r 2 ' + et^"-5 



2)" = - e/+t v/+Mt ri"+M2 r2"+ etc. 

 Let us also write for brevity 



W= [f] — t [7/] — JJ^ [mj — pi2 [m^] — etc. (552) 



(It will be observed that the value of W is entirely determined by 

 the nature of the physical system considered, and that the notion of 

 the dividing surface does not in any way enter into its definition.) 

 We shall then have 



0s = W+ v' ip' —p>"), (553) 



or, substituting for s and v' their values in terms of r, 



4 TT r2 0- = W + %7i r3 (;y - p"), (554) 



and eliminating by (550), 



%7tr^ {])' - p") = W, (555) 



/ 3W Y ^ ^ 



If we eliminate r instead 0, we have 



—r -, iFZT, = ^C (c'-5^) 



3(/-i/)- ' ^ ' 



/3TF(;y-;/)3\* , , 



Now, if we first suppose the difference of the pressures in the homo- 

 geneo\is masses to be very small, so that the surface of discontin- 

 uity is nearly plane, since without any important loss of generality 



