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

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

With this understanding, which is also to apply to $$\delta p$$ and $$\delta \sigma$$ 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 But it is evident that  where the second integral relates to the surfaces of discontinuity bounding the volume considered, and $$\delta N$$ denotes the normal component of the motion of an element of the surface, measured outward. Hence, Since this equation is true of each separate volume into which the system is divided, we may write for the whole system  where $$p'$$ and $$p''$$ denote the pressures on opposite sides of the element $$Ds$$, and $$\delta N$$ is measured toward the side specified by double accents.

Again, for each of the surfaces of discontinuity, taken separately, and  where $$c_{1}$$ and $$c_{2}$$ denote the principal curvatures of the surface (positive, when the centers are on the side opposite to that toward which $$\delta N$$ is measured), $$Dl$$ an element of the perimeter of the surface, and $$\delta T$$ 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 where the integration of the elements Dl extends to all the lines in which the surfaces of discontinuity meet, and the symbol $$\textstyle \sum$$ 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