Page:Scientific Papers of Josiah Willard Gibbs.djvu/365

These quantities will satisfy the following general relations:—

$d(\epsilon _{S})=td(\eta _{S})+\mu _{2}d(\Gamma _{2})+\mu _{3}d(\Gamma _{3})+{\text{etc.,}}$

(677)

$\varsigma =-(\eta _{S})dt-(\Gamma _{2})d\mu _{2}-(\Gamma _{3})d\mu _{3}={\text{etc.}}$

(678)

In strictness, these relations are subject to the same limitation as (674) and (675). But this limitation may generally be neglected. In fact, the values of $\varsigma ,(\epsilon _{S})$, etc. must in general be much less affected by variations in the state of strain of the surface of the solid than those of $\sigma \epsilon _{S(1)}$, etc.

The quantity $\varsigma$ evidently represents the tendency to contraction in that portion of the surface of the fluid which is in contact with the solid. It may be called the superficial tension of the fluid in contact with the solid. Its value may be either positive or negative.

It will be observed for the same solid surface and for the same temperature but for different fluids the values of $\sigma$ (in all cases to which the definition of this quantity is applicable) will differ from those of $\varsigma$ by a constant, viz., the value of $\sigma$ for the solid surface in a vacuum.

For the condition of equilibrium of two different fluids at a line on the surface of the solid, we may easily obtain

$\sigma _{AB}\cos \alpha =\varsigma _{BS}-\varsigma _{AS},$

(679)

the suffixes, etc., being used as in (672), and the condition being subject to the same modification when the fluids meet at an edge of the solid.

It must also be regarded as a condition of theoretical equilibrium at the line considered (subject, like (679), to limitation on account of passive resistances to motion), that if there are any foreign substances in the surfaces ${\text{A-S}}$ and ${\text{B-S}}$, the potentials for these substances shall have the same value on both sides of the line; or, if any such substance is found only on one side of the line, that the potential for that substance must not have a less value on the other side; and that the potentials for the components of the mass $A$, for example, must have the same values in the surface ${\text{B-C}}$ as in the mass $A$, or, if they are not actual components of the surface ${\text{B-C}}$, a value not less than in $A$. Hence, we cannot determine the difference of the surface-tensions of two fluids in contact with the same solid, by bringing them together upon the surface of the solid, unless these conditions are satisfied, as well as those which are necessary to prevent the mixing of the fluid masses.

The investigation on pages 276–282 of the conditions of equilibrium for a fluid system under the influence of gravity may easily be extended to the case in which the system is bounded by or includes solid masses, when these can be treated as rigid and incapable of