Page:Scientific Papers of Josiah Willard Gibbs.djvu/227

The six quantities $X_{X},Y_{Y},Z_{Z},X_{Y}$, or $Y_{X},Y_{Z}$ or $Z_{Y}$, and $Z_{X}$ or $X_{Z}$ are called the rectangular components of stress, the three first being the longitudinal stresses and the three last the shearing stresses. The mechanical conditions of internal equilibrium for a solid under the influence of gravity may therefore be expressed by the equations

${\frac {dX_{X}}{dx}}+{\frac {dX_{Y}}{dy}}+{\frac {dX_{Z}}{dz}}=0,$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}$ (377)
${\frac {dY_{X}}{dx}}+{\frac {dY_{Y}}{dy}}+{\frac {dY_{Z}}{dz}}=0,$
${\frac {dZ_{X}}{dx}}+{\frac {dZ_{Y}}{dy}}+{\frac {dZ_{Z}}{dz}}=g\Gamma ,$

where $\Gamma$ denotes the density of the element to which the other symbols relate. Equations (375), (376) are rather to be regarded as expressing necessary relations (when $X_{X},...Z_{Z}$ are regarded as internal forces determined by the state of strain of the solid) than as expressing conditions of equilibrium. They will hold true of a solid which is not in equilibrium,—of one, for example, through which vibrations are propagated,—which is not the case with equations (377).

Equation (373) expresses the mechanical conditions of equilibrium for a surface of discontinuity within the solid. If we set the coefficients of $\delta x,\delta y,\delta z$, separately equal to zero we obtain

$(\alpha 'X_{X'}+\beta 'X_{Y'}+\gamma 'X_{Z'})_{1}+(\alpha 'X_{X'}+\beta 'X_{Y'}+\gamma 'X_{Z'})_{2}=0,$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}$ (378)
$(\alpha 'Y_{X'}+\beta 'Y_{Y'}+\gamma 'Y_{Z'})_{1}+(\alpha 'Y_{X'}+\beta 'Y_{Y'}+\gamma 'Y_{Z'})_{2}=0,$
$(\alpha 'Z_{X'}+\beta 'Z_{Y'}+\gamma 'Z_{Z'})_{1}+(\alpha 'Z_{X'}+\beta 'Z_{Y'}+\gamma 'Z_{Z'})_{2}=0.$

Now when the $\alpha ',\beta ',\gamma '$ represent the direction-cosines of the normal in the state of reference on the positive side of any surface within the solid, an expression of the form

$\alpha 'X_{X'}+\beta 'X_{Y'}+\gamma 'X_{Z'}$

(379)

represents the component parallel to $X$ of the force exerted upon the surface in the strained state by the matter on the positive side per unit of area measured in the state of reference. This is evident from the consideration that in estimating the force upon any surface we may substitute for the given surface a broken one consisting of elements for each of which either $x'$ or $y'$ or $z'$ is constant. Applied to a surface bounding a solid, or any portion of a solid which may not be continuous with the rest, when the normal is drawn outward as usual, the same expression taken negatively represents the component parallel to $X$ of the force exerted upon the surface (per unit of surface measured in the state of reference) by the interior of the solid, or of the portion considered. Equations (378) therefore express the condition that the force exerted upon the surface of