These functions define the coexistence curve which is the locus of the saturated liquid and saturated vapor states of the van der Waals fluid. In Fig. 4 this curve is plotted in blue together with the spinodal curve in black, calculated from T = T ∗ ∗ ( v ~ ~ − − 1 ) 2 / v ~ ~ 3 p = p ∗ ∗ ( v ~ ~ − − 2 ) / v ~ ~ 3 where v ~ ~ = v / v ∗ ∗ is a parameter. The variables used in making these plots are the reduced (dimensionless) variables, p r = p / p c , v r = v / v c , and T r = T / T c where the c subscripted quantities are the critical point values. They are defined by, ∂ ∂ v p | T = 0 , and ∂ ∂ v 2 p | T = 0 at the critical point, and are measurable quantities. The relations p ∗ ∗ / p c = 27 , v ∗ ∗ / v c = 1 / 3 , T ∗ ∗ / T c = 27 / 8 are used to convert the star quantities in the solution to the c quantities used in the figures. The curve agrees completely with the numerical results referenced earlier. In the region inside the spinodal curve there are two states at each point, one stable and one metastable, either superheated liquid to the right of the blue curve, or subcooled vapor to the left, while outside the spinodal curve there is one stable state at each point. In Fig. 5 the region under the (dot dash black) spinodal curve contains no homogeneous stable states while between the (dot dash red) coexistence and spinodal curves there is one metastable state at every point, and outside the coexistence curve there is one stable state at each point. The two blue and two green circles denote the saturated liquid and vapor states on their respective isotherms. There are also observed heterogeneous states everywhere under the coexistence curve that satisfy the lever rule; however, they are not homogeneous states of the van der Waals equation, so their existence, indicated by horizontal lines connecting the saturation points on each subcritical isotherm, is not displayed. Also the abscissa in this figure is logarithmic, not linear, in order to show more of the vapor region at large v r without excessively compressing the liquid and unstable regions at small v r ; however, this device distorts areas, so the two areas I and II in Fig.1 would not appear equal here.