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

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.

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

Hence,

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

(634)

where

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

may be written for $[{\text{ABC}}]$, 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),

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

(635)

which with (634) gives

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

(636)

Since this is the only value of $p_{D}$ 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