Page:Scientific Papers of Josiah Willard Gibbs.djvu/316

in which ${\displaystyle {\frac {d\sigma }{d\omega _{1}}},{\frac {d\sigma }{d\omega _{2}}}}$ are determined by the function mentioned, and ${\displaystyle \delta \omega _{1},\delta \omega _{2}}$, by the component of the motion of ${\displaystyle Ds}$ which lies in the plane of the surface.

With this understanding, which is also to apply to ${\displaystyle \delta p}$ and ${\displaystyle \delta \sigma }$ when contained implicitly in any expression, we shall proceed to the reduction of the condition (606).

With respect to any one of the volumes into which the system is divided by the surfaces of discontinuity, we may write

${\displaystyle \int p\delta Dv=\delta \int pDv-\int \delta pDv.}$

But it is evident that

${\displaystyle \delta \int pDv=\int p\delta NDs,}$

where the second integral relates to the surfaces of discontinuity bounding the volume considered, and ${\displaystyle \delta N}$ denotes the normal component of the motion of an element of the surface, measured outward. Hence,

${\displaystyle \int p\delta Dv=\int p\delta NDs-\int \delta pDv.}$

Since this equation is true of each separate volume into which the system is divided, we may write for the whole system

${\displaystyle \int p\delta Dv=\int (p'-p'')\delta NDs-\int \delta pDv,}$

(609)

where ${\displaystyle p'}$ and ${\displaystyle p''}$ denote the pressures on opposite sides of the element ${\displaystyle Ds}$, and ${\displaystyle \delta N}$ is measured toward the side specified by double accents.

Again, for each of the surfaces of discontinuity, taken separately,

${\displaystyle \int \sigma \delta Ds=\delta \int \sigma Ds-\int \delta \sigma Ds,}$

and

${\displaystyle \delta \int \sigma Ds=\int \sigma (c_{1}+c_{2})\delta NDs+\int \sigma \delta TDl,}$

where ${\displaystyle c_{1}}$ and ${\displaystyle c_{2}}$ denote the principal curvatures of the surface (positive, when the centers are on the side opposite to that toward which ${\displaystyle \delta N}$ is measured), ${\displaystyle Dl}$ an element of the perimeter of the surface, and ${\displaystyle \delta T}$ the component of the motion of this element which lies in the plane of the surface and is perpendicular to the perimeter (positive, when it extends the surface). Hence we have for the whole system

${\displaystyle \int \sigma \delta Ds=\int \sigma (c_{1}+c_{2})\delta N+Ds+\int \textstyle \sum \displaystyle (\sigma \delta T)Dl-\int \delta \sigma Ds,}$

(610)

where the integration of the elements Dl extends to all the lines in which the surfaces of discontinuity meet, and the symbol ${\displaystyle \textstyle \sum }$ denotes a summation with respect to the several surfaces which meet in such a line.

By equations (609) and (610), the general condition of mechanical equilibrium is reduced to the form

${\displaystyle -\int (p'-p'')\delta N+\int \delta pDv+\int \sigma (c_{1}+c_{2})\delta NDs+\int \textstyle \sum \displaystyle (\sigma \delta T)Dl-\int \delta \sigma Ds+\int g\gamma \delta zDv+\int g\Gamma \delta zDs=0.}$