Page:Scientific Papers of Josiah Willard Gibbs.djvu/366

dissolution. The general condition of mechanical equilibrium would be of the form

$-\int p\delta Dv+\int g\gamma \,\,\delta z\,\,Dv+\int \sigma \,\,\delta Ds+\int g\Gamma \,\,\delta z\,\,Ds+\int g\,\,\delta {\text{z}}\,\,Dm+\int \varsigma \,\,\delta Ds+\int g(\Gamma )\delta z\,\,Ds=0,$

(680)

where the first four integrals relate to the fluid masses and the surfaces which divide them, and have the same signification as in equation (606), the fifth integral relates to the movable solid masses, and the sixth and seventh to the surfaces between the solids and fluids, $(\Gamma )$ denoting the sum of the quantities $(\Gamma _{2}),(\Gamma _{3})$, etc. It should be observed that at the surface where a fluid meets a solid $\delta {\text{z}}$ and $\delta z$, which indicate respectively the displacements of the solid and the fluid, may have different values, but the components of these displacements which are normal to the surface must be equal.

From this equation, among other particular conditions of equilibrium, we may derive the following:—

$d\varsigma =g(\Gamma )dz$

(681)

(compare (614)), which expresses the law governing the distribution of a thin fluid film on the surface of a solid, when there are no passive resistances to its motion.

By applying equation (680) to the case of a vertical cylindrical tube containing two different fluids, we may easily obtain the well-known theorem that the product of the perimeter of the internal surface by the difference $\varsigma '-\varsigma ''$ of the superficial tensions of the upper and lower fluids in contact with the tube is equal to the excess of weight of the matter in the tube above that which would be there, if the boundary between the fluids were in the horizontal plane at which their pressures would be equal. In this theorem, we may either include or exclude the weight of a film of fluid matter adhering to the tube. The proposition is usually applied to the column of fluid in mass between the horizontal plane for which $p'=p''$ and the actual boundary between the two fluids. The superficial tensions $\varsigma '$ and $\varsigma ''$ are then to be measured in the vicinity of this column. But we may also include the weight of a film adhering to the internal surface of the tube. For example, in the case of water in equilibrium with its own vapor in a tube, the weight of all the water-substance in the tube above the plane $p'=p''$, diminished by that of the water-vapor which would fill the same space, is equal to the perimeter multiplied by the difference in the values of $\varsigma$ at the top of the tube and at the plane $p'=p''$. If the height of the tube is infinite, the value of $\varsigma$ at the top vanishes, and the weight of the film of water adhering to the tube and of the mass of liquid water above the plane $p'=p''$ diminished by the weight of vapor which would fill the same space is equal in numerical value but of opposite sign to the product of the perimeter of the internal