Page:Scientific Papers of Josiah Willard Gibbs.djvu/335

that the surface ${\text{abc}}$ is the surface ${\text{D-E}}$ and therefore perpendicular to $\delta \epsilon$, etc. Let ${\text{tetr abcd}}$, ${\text{trian abc}}$, etc. denote the volume of the tetrahedron or the area of the triangle specified, $\sin {({\text{ab, bc}})},\sin {({\text{abc, dbc}})},\sin {({\text{abc, ad}})}$, etc. the sines of the angles made by the lines and surfaces specified, and $[{\text{BCDE}}],[{\text{CDEA}}]$, etc. the volumes of tetrahedra having edges equal to the tensions of the surfaces between the masses specified. Then, since we may express the volume of a tetrahedron either by ${\tfrac {1}{3}}$ of the product of one side, an edge leading to the opposite vertex, and the sine of the angle which these make, or by ${\tfrac {2}{3}}$ of the product of two sides divided by the common edge and multiplied by the sine of the included angle,

$v_{A}:v_{B}$	$::{\text{tetr bcde}}:{\text{tetr acde}}$
	$::{\text{bc}}\,\,\sin {({\text{ab, bc}})}:{\text{ac}}\,\,\sin {({\text{ac, cde}})}$
	$::\sin {({\text{ba, ac}})}\sin {({\text{bc, cde}})}:{\text{ac}}\,\,\sin {({\text{ac, cde}})}$
	$::\sin {(\gamma \delta \epsilon ,\,\beta \delta \epsilon )}\sin {(\alpha \delta \epsilon ,\,\alpha \beta )}:\,\sin {(\gamma \delta \epsilon ,\,\alpha \delta \epsilon )}\,\sin {(\beta \delta \epsilon ,\,\alpha \beta )}$
	$::{\frac {{\text{tetr }}\alpha \beta \delta \epsilon \,{\text{tetr }}\beta \alpha \delta \epsilon }{{\text{trian }}\beta \delta \epsilon \,{\text{trian }}\alpha \delta \epsilon }}\,\,:\,\,{\frac {{\text{tetr }}\gamma \alpha \delta \epsilon \,{\text{tetr }}\alpha \beta \delta \epsilon }{{\text{trian }}\alpha \delta \epsilon \,{\text{trian }}\beta \delta \epsilon }}$
	$::{\text{tetr }}\gamma \beta \delta \epsilon \,\,:\,\,{\text{tetr }}\gamma \alpha \delta \epsilon$
	$::[{\text{BCDE}}]\,\,:\,\,[{\text{CDEA}}].$

Hence,

$v_{A}:v_{B}:v_{C}:v_{D}::[{\text{BCDE}}]:[{\text{CDEA}}]:[{\text{DEAB}}]:[{\text{EABC}}],$

(641)

and (640) may be written

$p_{E}={\frac {[{\text{BCDE}}]p_{A}+[{\text{CDEA}}]p_{B}+[{\text{DEAB}}]p_{C}+[{\text{EABC}}]p_{D}}{[{\text{BCDE}}]+[{\text{CDEA}}]+[{\text{DEAB}}]+[{\text{EABC}}]}}\cdot$

(642)

If the value of $p_{E}$ is less than this, when the tensions satisfy the critical relation, the point where vertices of the masses $A,B,C,D$ meet is stable with respect to the formation of any mass of the nature of $E$. But if the value of $p_{E}$ is greater, either the masses $A,B,C,D$ cannot meet at a point in equilibrium, or the equilibrium will be at least practically unstable.

When the tensions of the new surfaces are too small to satisfy the critical relation with the other tensions, these surfaces will be convex toward $E$; when their tensions are too great for that relation, the surfaces will be concave toward $E$. In the first case, $W_{V}$ is negative, and the equilibrium of the five masses $A,B,C,D,E$ is stable, but the equilibrium of the four masses $A,B,C,D$ meeting at a point is impossible or at least practically unstable. This is subject to the limitation that when $p_{E}$ is sufficiently small the mass $E$ which will form will be so small that it may be neglected. This will only be the case when $p_{E}$ is smaller—in general considerably smaller—than the second member of (642). In the second case, the equilibrium of the five masses $A,B,C,D,E$ will be unstable, but the equilibrium