Gibbs' geometrical method for determining coexisting states is illustrated by the transparent tangent plane that straddles the (instability-induced) saddle curvature in the red region of the surface. The plane's two points of tangency (marked with diamond cutouts) share the same principal slopes and project to the same intercept on the U- axis (where the like-colored chip cuts the checkered pole). In these mass-scaled coordinates this geometrical situation confirms the equality of temperature, pressure, and chemical potential between the two tangent states and was shown by Gibbs to be consistent with entropy maximization in an isolated system.

As the tangent plane rolls to the left or right of the position shown, this geometrical situation persists, and the moving tangencies describe the two branches of the vapor-liquid coexistence curve (the blue-yellow boundary). A more complete model would show the total USV surface, with the analogous regions of liquid-solid and vapor-solid equilibrium and also the triple point - where the rolling plane finds three tangent points and locks rigidly onto the surface. Such a model was constructed by Clark and Katz and may be seen in Transactions of the Royal Society of Canada, 3rd series, Sec. III, No. 33 (1939), p. 59.

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