Page:Scientific Papers of Josiah Willard Gibbs.djvu/354

318 the solid in the two masses—the same condition which would subsist if both masses were fluid.

Moreover, the compressibility of all solids is so small that, although or may not represent the true tension of the surface, nor $$p + (c_{1} + c_{2})\sigma$$ the true pressure in the solid when its stresses are isotropic, the quantities $$\epsilon_{V}'$$ and $$\eta_{V}'$$ if calculated for the pressure $$p + (c_{1} + c_{2})\sigma$$ with the actual temperature will have sensibly the same values as if calculated for the true pressure of the solid. Hence, the second member of equation (661), when the stresses of the solid are sensibly isotropic, is sensibly equal to the potential of the same body at the same temperature but with the pressure $$p'' + (c_{1} + c_{2})\sigma$$ and the condition of equilibrium with respect to dissolving for a solid of isotropic stresses may be expressed with sufficient accuracy by saying that the potential for the substance of the solid in the fluid must have this value. In like manner, when the solid is not in a state of isotropic stress, the difference of the two pressures in question will not sensibly affect the values of $$\epsilon_{V}'$$ and $$\eta_{V}'$$, and the value of the second member of the equation may be calculated as if $$p'' + (c_{1} + c_{2})\sigma$$ represented the true pressure in the solid in the direction of the normal to the surface. Therefore, if we had taken for granted that the quantity or represents the tension of a surface between a solid and a fluid, as it does when both masses are fluid, this assumption would not have led us into any practical error in determining the value of the potential $$\mu_{1}''$$ which is necessary for equilibrium. On the other hand, if in the case of any amorphous body the value of or differs notably from the true surface-tension, the latter quantity substituted for $$\sigma$$ in (661) will make the second member of the equation equal to the true value of $$\mu_{1}'$$, when the stresses are isotropic, but this will not be equal to the value of $$\mu_{1}''$$ in case of equilibrium, unless $$c_{1} + c_{2} = 0$$.

When the stresses in the solid are not isotropic, equation (661) may be regarded as expressing the condition of equilibrium with respect to the dissolving of the solid, and is to be distinguished from the condition of equilibrium with respect to an increase of solid matter, since the new matter would doubtless be deposited in a state of isotropic stress. (The case would of course be different with crystalline bodies, which are not considered here.) The value of $$\mu_{1}''$$ necessary for equilibrium with respect to the formation of new matter is a little less than that necessary for equilibrium with respect to the dissolving of the solid. In regard to the actual behavior of the solid and fluid, all that the theory enables us to predict with certainty is that the solid will not dissolve if the value of the potential $$\mu_{1}''$$ is greater than that given by the equation for the solid with its distorting stresses, and that new matter will not be formed if the value of $$\mu_{1}''$$ is less than the same equation would give for the case of