Page:Scientific Papers of Josiah Willard Gibbs.djvu/230

energy, of entropy, and of the several component substances in the variable state of the solid. We shall obtain, on dividing the equation by $v_{V'}$,

$\epsilon _{V}+t\eta _{V}+p=\textstyle \sum _{1}\displaystyle (\mu _{1}\Gamma _{1}).$

(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 $\mu _{1}$ to denote the potential for this substance in the fluid, and $\Gamma$ to denote the density of the solid in the variable state ($\Gamma '$, as before denoting its density in the state of reference), we shall have

$\epsilon _{V'}+t\eta _{V'}+pv_{V'}=\mu _{1}\Gamma ',$

(386)

and$\epsilon _{V}+t\eta _{V}+p=\mu _{1}\Gamma .$

(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 $\epsilon ,\eta ,v,$ and $m$, its energy, entropy, volume, and mass, we have

$\epsilon -t\eta +pv=\mu _{1}m.$

(388)

Now the mechanical conditions of equilibrium for the surface where a solid meets a fluid require that the traction upon the surface determined 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 $\alpha ,\beta ,\gamma$ (compare (379), and for the notation $X_{X}$, etc., page 190),

$S=$	$\alpha (\alpha X_{X}+\beta X_{Y}+\gamma X_{Z})$
	$+\beta (\alpha Y_{X}+\beta Y_{Y}+\gamma Y_{Z})$
	$+\gamma (\alpha Z_{X}+\beta Z_{Y}+\gamma Z_{Z}),$

or, by (375), (376),

$S=\alpha ^{2}X_{X}+\beta ^{2}Y_{X}+\gamma ^{2}Z_{X}+2\alpha \beta X_{Y}+2\beta \gamma Y_{Z}+2\gamma \alpha Z_{X}.$

(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