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

when ${\displaystyle v_{D}}$ 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 ${\displaystyle v_{A},v_{B},v_{C},v_{D}}$, for sufficiently small values of ${\displaystyle v_{D}}$, are determined by the tensions alone, and for convenience in calculating these ratios, we may suppose ${\displaystyle p_{A},p_{B},p_{C}}$ to be equal, which will make the figure ${\displaystyle abcd}$ absolutely rectilinear, and make ${\displaystyle p_{D}}$ 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 ${\displaystyle v_{A},v_{B},v_{C},v_{D}}$ in terms of the tensions in the following manner. We shall write ${\displaystyle [{\text{DBC}}],[{\text{DCA}}]}$, etc., to denote the areas of triangles having sides equal to the tensions of the surfaces between the masses specified.

${\displaystyle v_{A}:V_{B}}$	${\displaystyle ::{\text{triangle}}\,\,bdc:{\text{triangle}}\,\,adc}$
	${\displaystyle ::bc\sin bcd:ac\sin acd}$
	${\displaystyle ::\sin bac\sin bcd:\sin abc\sin acd}$
	${\displaystyle ::\sin \gamma \delta \beta \sin \delta \alpha \beta :\sin \gamma \delta \alpha \sin \delta \beta \alpha }$
	${\displaystyle ::\sin {\gamma \delta \beta }\,\,\delta \beta :\sin {\gamma \delta \alpha }\,\,\delta \alpha }$
	${\displaystyle ::{\text{triangle}}\,\,\gamma \delta \beta :{\text{triangle}}\,\,\gamma \delta \alpha }$
	${\displaystyle ::[{\text{DBC}}]:[{\text{DCA}}].}$

Hence,

${\displaystyle v_{A}:v_{B}:v_{C}:v_{D}::[{\text{DBC}}]:[{\text{DCA}}]:[{\text{DAB}}]:[{\text{ABC}}],}$

(634)

where

${\displaystyle {\tfrac {1}{4}}{\sqrt {}}[(\sigma _{AB}+\sigma _{BC}+\sigma _{CA})(\sigma _{AB}+\sigma _{BC}-\sigma _{CA})(\sigma _{BC}+\sigma _{CA}-\sigma _{AB})(\sigma _{CA}+\sigma _{AB}-\sigma _{BC})]}$

may be written for ${\displaystyle [{\text{ABC}}]}$, and analogous expressions for the other symbols, the sign ${\displaystyle {\sqrt {}}}$ 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 ${\displaystyle v_{D}}$ is sufficiently small. Now if ${\displaystyle p_{A}=p_{B}=p_{C},}$ ${\displaystyle p_{D}}$ will have the same value, and we shall have by (627) ${\displaystyle W_{V}=0}$, and by (633) ${\displaystyle W_{S}=0}$. But when ${\displaystyle v_{D}}$ is very small, the value of ${\displaystyle W_{S}}$ is entirely determined by the tensions and ${\displaystyle v_{D}}$. Therefore, whenever the tensions satisfy the condition supposed, and ${\displaystyle v_{D}}$ is very small (whether ${\displaystyle p_{A},p_{B},p_{C}}$ are equal or unequal),

${\displaystyle 0=W_{S}=W_{V}=p_{D}v_{D}-p_{A}v_{A}-p_{B}v_{B}-p_{C}v_{C}.}$

(635)

which with (634) gives

${\displaystyle p_{D}={\frac {[{\text{DBC}}]p_{A}+[{\text{DCA}}]p_{B}+[{\text{DAB}}]p_{C}}{[{\text{DBC}}]+[{\text{DCA}}]+[{\text{DAB}}]}}\cdot }$

(636)

Since this is the only value of ${\displaystyle p_{D}}$ for which equilibrium is possible when the tensions satisfy the condition supposed and ${\displaystyle v_{D}}$ is small, it follows that when ${\displaystyle p_{D}}$ has a less value, the line where the fluids