486 J. W. Gibbs — EquUibrinrn of TIeterof/eueoxs Substances. 



we add the above expression for the increment of energy near the 

 surface, we obtain the increment of energy for the whole system. 

 Now by (93) we have 



p" = - 6/ + t W + /./' r/ + f,, y/ + etc. 



By this equation and (659), our expression for the total increment of 

 energy in the system may be reduced to the form 



/\£y' — t ;/v' — /</' r,'+p" 4- (Ci+Cg) 0-] dJVDs. (660) 



In oi-der that this shall vanish for any values of SN^ it is necessary 

 that the coefficient of dJVDs shall vanish. This gives for the condi- 

 tion of equilibrium 



y I 



This equation is identical with (387), wdth the exception of the term 

 containing o", which vanishes when the surface is plane.* 



We may also observe that when the solid has no stresses except an 

 isotropic pressure, if the quantity represented by is equal to the true 

 tension of the surface, p" + [c^ -f- c.^) o' will represent the pressure in 

 the interior of the solid, and the second member of the equation Avill 

 represent [see equation (93)] the Aalue of the potential in the solid 

 for the substance of which it consists. In this case, therefore, the 

 equation reduces to 



that is, it expresses the equality of the potentials for the substance of 

 the solid in the two masses — the same condition which Avould subsist 

 if both masses were fluid. 



Moreover, the compressibility of all solids is so small that, althoixgh 

 o' may not represent the true tension of the surface, wov p"-{- (cj +^2)0" 

 the true pressure in the solid when its stresses are isotropic, the quan- 

 tities fy' and 7a' if calculated for tlie pressure p" -j- (cj -j-^'g) ^ with 

 the actual temperature will have sensibly the same values as if calcu- 

 lated for the true pressure of the solid. Hence, the second member 



relating to the solid mass (when this is in a state of isotropic stress), which, as we 

 shall see, may not always have the same value. The other potentials n^, etc., have 

 the same values as in (659), and consist of two classes, one of which relates to sub- 

 stances which are components of the fluid mass, (these might be marked by the double 

 accents.) and the other relates to substances found only at the surface of discontinuity. 

 The expressions to be multiplied l)y the potentials of this latter class all have the 

 value zero. 



* In equation (38V), the density of the solid is denoted by F, which is therefore 

 equivalent to y/ in (661). 



