194 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



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

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

 by v v ,, 



e v -^v+^ = 2: i (^ i r i ). (385) 



It will be remembered that the summation relates to the several 

 components 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 fa to denote the potential for this 

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

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

 reference), we shall have 



T -t^+pv T = jui i r t (386) 



and e v tij v +p = faT. (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 e, TJ, v, and ra, its energy, entropy, volume, 



and mass, we have 



tij +pv = fam. (388) 



Now the mechanical conditions of equilibrium for the surface where 

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

 mined by the state of strain of the solid shall be normal to the surface. 

 This condition is always satisfied with respect to three surfaces at 

 right angles to one another. In proving this well-known proposition, 

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

 which is arbitrary, coincident with the state under discussion, 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, /3, y (compare (379), and for the notation X x , etc., 

 page 190), 



or, by (375), (376), 



(389) 



We may also choose any convenient directions for the co-ordinate 

 axes. Let us suppose that the direction of the axis of X is so chosen 

 that the value of S for the surface perpendicular to this axis is as 

 great as for any other surface, and that the direction of the axis of Y 

 (supposed at right angles to X) is such that the value of S for the 



