Page:Scientific Papers of Josiah Willard Gibbs.djvu/315

It will be observed that this condition has the same form as if the different fluids were separated by heavy and elastic membranes without rigidity and having at every point a tension uniform in all directions in the plane of the surface. The variations in this formula, beside their necessary geometrical relations, are subject to the conditions that the external surface of the system, and the lines in which the surfaces of discontinuity meet it, are fixed. The formula may be reduced by any of the usual methods, so as to give the particular conditions of mechanical equilibrium. Perhaps the following method will lead as directly as any to the desired result.

It will be observed the quantities affected by $\delta$ in (606) relate exclusively to the position and size of the elements of volume and surface into which the system is divided, and that the variations $\delta p$ and $\delta \sigma$ do not enter into the formula either explicitly or implicitly. The equations of condition which concern this formula also relate exclusively to the variations of the system of geometrical elements, and do not contain either $\delta p$ or $\delta \sigma$. Hence, in determining whether the first member of the formula has the value zero for every possible variation of the system of geometrical elements, we may assign to $\delta p$ and $\delta \sigma$ any values whatever which may simplify the solution of the problem, without inquiring whether such values are physically possible.

Now when the system is in its initial state, the pressure $p$, in each of the parts into which the system is divided by the surfaces of tension, is a function of the co-ordinates which determine the position of the element $Dv$, to which the pressure relates. In the varied state of the system, the element $Dv$ will in general have a different position. Let the variation $\delta p$ be determined solely by the change in position of the element $Dv$. This may be expressed by the equation

$\delta p={\frac {dp}{dx}}\delta x+{\frac {dp}{dy}}\delta y+{\frac {dp}{dz}}\delta z,$

(607)

in which ${\frac {dp}{dx}},{\frac {dp}{dy}},{\frac {dp}{dz}}$ are determined by the function mentioned, and $\delta x,\delta y,\delta z$ by the variation of the position of the element $Dv$.

Again, in the initial state of the system the tension $\sigma$, in each of the different surfaces of discontinuity, is a function of two co-ordinates $\omega _{1},\omega _{2}$, which determine the position of the element $Ds$. In the varied state of the system, this element will in general have a different position. The change of position may be resolved into a component lying in the surface and another normal to it. Let the variation $\delta \sigma$ be determined solely by the first of these components of the motion of $Ds$. This may be expressed by the equation

$\delta \sigma ={\frac {d\sigma }{d\omega _{1}}}\delta \omega _{1}+{\frac {d\sigma }{d\omega _{2}}}\delta \omega _{2},$

(608)