J. W. Gibhs — Eijullihrliiya of Heterogeneous f>}(hstanees. 353 



jKnients (see page 117) of the fluiil, 8N' is incapable of positive 

 values, as the quantity of the solid cannot be increased, and it is 

 sufficient for equilil>rium that 



To express condition (383) in a form independent of the state of 

 reference, we may use fy, '/v, ^\, ete-, to denote the densities of 

 energy, of entropy, and of the several component substances in the 

 variable state of the solid. We shall obtain, on dividing the equa- 

 tion by Uv<, 



fv-«//v+7>=^,(;^J^i). (-^85) 



It will be remembered that tlie summation relates to the several 

 corajjonents of the solid. If the solid is of uniform composition 

 throughout, or if we only care to consider the contact of the solid 

 and the fluid at a single point, we may treat the solid as composed of 

 a single substance. If we use yWj to denote the potential for this 

 substance in the fluid, and T to denote the density of the solid in the 

 variable state, (/"', as before denoting its density in the state of 

 reference,) we shall have 



fv» - «?/v, +i^ yv-= /'] ^^', (=^-^«) 



and 



fv — «//v+^> = /'i^^- (387) 



To fix our ideas in discussing this condition, let us apply it to the 

 case of a solid body which is homogeneous in nature and in state of 

 strain. If we denote by £, //, y, and /«, its energy, entropy, volume, 

 and mass, we have 



s — t)] -\- })V := u.^ m. (388) 



Now the mechanical conditions of equilibrium for the surface where 

 a solid meets a fluid reqiiire that the traction upon the surface deter- 

 mined by the state of strain of the solid shall be normal to the sur- 

 face. This condition is always satisfied with respect to three surfaces 

 at right angles to one another. In proving this well known proposi- 

 tion, we shall lose nothing in generality, if we make the state of 

 reference, which is arbitrary, coincident Avith the state under discus- 

 sion, the axes to which these states are referred being also coincident. 

 We shall then have, for the normal component of the traction per unit 

 of surface across any surface for which the direction-cosines of the 

 normal are a, p, y, [compare (3 79), and for the notation X,, etc., 

 page 349,] 



