Page:Scientific Papers of Josiah Willard Gibbs.djvu/317

Arranging and combining terms, we have

$\int (g\gamma +dp)Dv+\int [(p''-p')\delta N+\sigma (c_{1}+c_{2})\delta N+g\Gamma \delta z-\delta \sigma ]Ds+\int \textstyle \sum \displaystyle (\sigma \delta T)Dl=0.$

(611)

To satisfy this condition, it is evidently necessary that the coefficients of $Dv,Ds$, and $Dl$ shall vanish throughout the system.

In order that the coefficient of $Dv$ shall vanish, it is necessary and sufficient that in each of the masses into which the system is divided by the surfaces of tension, $p$ shall be a function of $z$ alone, such that

${\frac {dp}{dz}}=-g\gamma .$

(612)

In order that the coefficient of $Ds$ shall vanish in all cases, it is necessary and sufficient that it shall vanish for normal and for tangential movements of the surface. For normal movements we may write

$\delta \sigma =0,$and$\delta z=\cos \theta \delta N,$

where $\theta$ denotes the angle which the normal makes with a vertical line. The first condition therefore gives the equation

$p'-p''=\sigma (c_{1}+c_{2})+g\Gamma \cos \theta ,$

(613)

which must hold true at every point in every surface of discontinuity. The condition with respect to tangential movements shows that in each surface of tension $\sigma$ is a function of $z$ alone, such that

${\frac {d\sigma }{dz}}=g\Gamma .$

(614)

In order that the coefficient of $Dl$ in (611) shall vanish, we must have, for every point in every line in which surfaces of discontinuity meet, and for any infinitesimal displacement of the line,

$\textstyle \sum \displaystyle (\sigma \delta T)=0.$

(615)

This condition evidently expresses the same relations between the tensions of the surfaces meeting in the line and the directions of perpendiculars to the line drawn in the planes of the various surfaces, which hold for the magnitudes and directions of forces in equilibrium in a plane.

In condition (603), the variations which relate to any component are to be regarded as having the value zero in any part of the system in which that substance is not an actual component.^[1] The same is true

1. The term actual component has been defined for homogeneous masses on page 64, and the definition may be extended to surfaces of discontinuity. It will be observed that if a substance is an actual component of either of the masses separated by a surface of discontinuity, it must be regarded as an actual component for that surface, as well as when it occurs at the surface but not in either of the contiguous masses.