Page:Scientific Papers of Josiah Willard Gibbs.djvu/330

294 when $$v_{D}$$ is sufficiently small the figure abed may be regarded as rectilinear, and that its angles are entirely determined by its tensions. Hence the ratios of $$v_{A}, v_{B}, v_{C}, v_{D}$$, for sufficiently small values of $$v_{D}$$, are determined by the tensions alone, and for convenience in calculating these ratios, we may suppose $$p_{A}, p_{B}, p_{C}$$ to be equal, which will make the figure $$abcd$$ absolutely rectilinear, and make $$p_{D}$$ equal to the other pressures, since it is supposed that this quantity has the value necessary for equilibrium. We may obtain a simple expression for the ratios of $$v_{A}, v_{B}, v_{C}, v_{D}$$ in terms of the tensions in the following manner. We shall write $$[\text{DBC}], [\text{DCA}]$$, etc., to denote the areas of triangles having sides equal to the tensions of the surfaces between the masses specified.

Hence, where  may be written for $$[\text{ABC}]$$, and analogous expressions for the other symbols, the sign $$\sqrt{}$$ denoting the positive root of the necessarily positive expression which follows. This proportion will hold true in any case of equilibrium, when the tensions satisfy the condition mentioned and $$v_{D}$$ is sufficiently small. Now if $$p_{A} = p_{B} = p_{C},$$ $$p_{D}$$ will have the same value, and we shall have by (627) $$W_{V} = 0$$, and by (633) $$W_{S} = 0$$. But when $$v_{D}$$ is very small, the value of $$W_{S}$$ is entirely determined by the tensions and $$v_{D}$$. Therefore, whenever the tensions satisfy the condition supposed, and $$v_{D}$$ is very small (whether $$p_{A}, p_{B}, p_{C}$$ are equal or unequal), which with (634) gives  Since this is the only value of $$p_{D}$$ for which equilibrium is possible when the tensions satisfy the condition supposed and $$v_{D}$$ is small, it follows that when $$p_{D}$$ has a less value, the line where the fluids