Page:Scientific Papers of Josiah Willard Gibbs.djvu/225

We may therefore use (361) to eliminate the fourth and fifth integrals from (360). If we multiply (362) by $p$, and take the integrals for the whole surface of the solid and for the fluid in contact with it, we obtain the equation {{MathForm2|(365)|$\int ^{F}p\delta Dv=-\int p(\alpha \delta x+\beta \delta y+\gamma \delta z)Ds-\int pv_{V'}\delta N'Ds',$ by means of which we may eliminate the sixth integral from (360). If we add equations (363) multiplied respectively by $\mu _{1},\mu _{2}$, etc., and take the integrals, we obtain the equation

$\textstyle \sum _{1}\displaystyle \int ^{F}\mu _{1}\delta Dm_{1}=-\int \textstyle \sum _{1}\displaystyle (\mu _{1}\Gamma _{1}')\delta N'Ds',$

(366)

by means of which we may eliminate the last integral from (360).

The condition of equilibrium is thus reduced to the form

$\iiint \textstyle \sum \sum '\displaystyle \left(X_{X'}\delta {\frac {dx}{dx'}}\right)dx'dy'dz'+\iiint g\gamma '\delta zdx'dy'dz'+\int \epsilon _{V'}\delta N'Ds'-\int t\eta _{V'}\delta N'Ds'+\int p(\alpha \delta x+\beta \delta y+\gamma \delta z)Ds+\int pv_{V'}\delta N'Ds'-\int \textstyle \sum _{1}\displaystyle (\mu _{1}\Gamma _{1}')\delta N'Ds'\geqq 0,$

(367)

in which the variations are independent of the equations of condition, and in which the only quantities relating to the fluids are $p$ and $\mu _{1},\mu _{2}$, etc.

Now by the ordinary method of the calculus of variations, if we write $\alpha ',\beta ',\gamma ',$ for the direction cosines of the normal to the surface of the solid in the state of reference, we have

$\iiint X_{X'}\delta {\frac {dx}{dx'}}dx'dy'dz'=\int \alpha 'X_{X'}\delta xDs'-\iiint {\frac {dX_{X'}}{dx'}}\delta xdx'dy'dz',$

(368)

with similar expressions for the other parts into which the first integral in (367) may be divided. The condition of equilibrium is thus reduced to the form

$-\iiint \textstyle \sum \sum '\displaystyle \left({\frac {dX_{X'}}{dx'}}\delta x\right)dx'dy'dz'+\iiint g\gamma '\delta zdx'dy'dz'+\int \textstyle \sum \sum '\displaystyle (\alpha 'X_{X'}\delta x)Ds'+\int p\textstyle \sum \displaystyle (\alpha \delta x)Ds+\int [\epsilon _{V'}-t\eta _{V'}+pv_{V'}-\textstyle \sum _{1}\displaystyle (\mu _{1}\Gamma _{1}')]\delta N'Ds'\geqq 0.$

(369)

It must be observed that if the solid mass is not continuous throughout in nature and state, the surface-integral in (368), and therefore the first surface-integral in (369), must be taken to apply not only to the external surface of the solid, but also to every surface of discontinuity within it, and that with reference to each of the two masses separated by the surface. To satisfy the condition of