Page:Scientific Papers of Josiah Willard Gibbs.djvu/226

equilibrium, as thus understood, it is necessary and sufficient that throughout the solid mass

$\textstyle \sum \sum '\displaystyle \left({\frac {dX_{X'}}{dx'}}\delta x\right)-g\Gamma '\delta z=0;$

(370)

that throughout the surfaces where the solid meets the fluid

$Ds'\textstyle \sum \sum '\displaystyle (\alpha 'X_{X'}\delta x)+Dsp\textstyle \sum \displaystyle (\alpha \delta x)=0,$

(371)

and$[\epsilon _{V'}-t\eta _{V'}+pv_{V'}-\textstyle \sum _{1}\displaystyle (\mu _{1}\Gamma _{1}')]\delta N'\geqq 0;$

(372)

and that throughout the internal surfaces of discontinuity where the suffixed numerals distinguish the expressions relating to the masses on opposite sides of a surface of discontinuity.

Equation (370) expresses the mechanical conditions of internal equilibrium for a continuous solid under the influence of gravity. If we expand the first term, and set the coefficients of $\delta x,\delta y$, and $\delta z$ separately equal to zero, we obtain

${\frac {dX_{X'}}{dx'}}+{\frac {dX_{Y'}}{dy'}}+{\frac {dX_{Z'}}{dz'}}=0,$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}$ (374)
${\frac {dY_{X'}}{dx'}}+{\frac {dY_{Y'}}{dy'}}+{\frac {dY_{Z'}}{dz'}}=0,$
${\frac {dZ_{X'}}{dx'}}+{\frac {dZ_{Y'}}{dy'}}+{\frac {dZ_{Z'}}{dz'}}=g\Gamma ',$

The first member of any one of these equations multiplied by $dx'dy'dz'$ evidently represents the sum of the components parallel to one of the axes $X,Y,Z$ of the forces exerted on the six faces of the element $dx'dy'dz'$ by the neighboring elements.

As the state which we have called the state of reference is arbitrary, it may be convenient for some purposes to make it coincide with the state to which $x,y,z$ relate, and the axes $X',Y',Z'$ with the axes $X,Y,Z$. The values of $X_{X'},...Z_{Z'}$ on this particular supposition may be represented by the symbols $X_{X},...Z_{Z}$. Since

$X_{Y'}={\frac {d\epsilon _{V'}}{d{\frac {dx}{dy'}}}},$and$Y_{X'}={\frac {d\epsilon _{V'}}{d{\frac {dy}{dx'}}}},$

and since, when the states, $x,y,z$ and $x',y',z'$ coincide, and the axes $X,Y,Z,$ and $X',Y',Z',d{\frac {dx}{dy'}}$, and $d{\frac {dy}{dx'}}$, represent displacements which differ only by a rotation, we must have

$X_{Y}=Y_{X},$

(375)

and for similar reasons,

$Y_{Z}=Z_{Y},$$Z_{X}=X_{Z}.$

(376)