Here the integral has been split into three parts using the three real roots of the cubic corresponding to the liquid, v l , unstable, v u , and vapor, v v , states respectively. These integrals can best be visualized by viewing Fig. 1 rotated 90 ∘ ∘ counterclockwise in the paper plane then 180 ∘ ∘ about the v axis so that v appears on the leftside ordinate of the curve as shown in the accompanying graph. In this view the function v ( p , T ) clearly is multi-valued; this is the reason it requires three real functions to describe its behavior between p m i n and p m a x . Now on splitting the middle integral into two g g − − g f = ∫ ∫ p s p m i n v l d p + ∫ ∫ p m i n p s v u d p + ∫ ∫ p s p m a x v u d p + ∫ ∫ p m a x p s v v d p = 0 The first two integrals here are area I while the second two are the negative of area II. The two areas add to zero hence their magnitudes are equal according to this Gibbs criterion. This is again the equal area rule of Maxwell, the Maxwell construction, and it can also be shown analytically. Since d ( p v ) = p d v + v d p , d g = v d p − − s d T = − − p d v + d ( p v ) − − s d T . Integrating this for constant temperature from state f to g with the Gibbs condition produces g g − − g f = − − ∫ ∫ v f v g p ( v , T ) d v + p s ( v g − − v f ) = 0 which is Maxwell's result. This equal area rule can also be derived by making use of the Helmholtz free energy. In any event the Maxwell construction derives from the Gibbs condition of material equilibrium. However, even though g f = g g is more fundamental it is more abstract than the equal area rule, which is understood geometrically.