For a binary mixture the Euler equation, can be written in the form f = − − p v + μ μ 1 x 1 + μ μ 2 x 2 = − − p v + ( μ μ 2 − − μ μ 1 ) x + μ μ 1 Here μ μ j = ∂ ∂ x j f are the molar chemical potentials of each substance, j = 1 , 2 . For p , μ μ 1 and μ μ 2 , all constant this is the equation of a plane with slopes − − p in the v direction, μ μ 2 − − μ μ 1 in the x direction, and intercept μ μ 1 . As in the case of a single substance, here the plane and the surface can have a double tangent and the locus of the coexisting phase points forms a curve on each surface. The coexistence conditions are that the two phases have the same T , p , μ μ 2 − − μ μ 1 , and μ μ 1 ; the last two are equivalent to having the same μ μ 1 and μ μ 2 individually, which are just the Gibbs conditions for material equilibrium in this situation. Although this case is similar to the previous one of a single component, here the geometry can be much more complex. The surface can develop a wave (called a plait or fold in the literature) in the x direction as well as the one in the v direction. Therefore, there can be two liquid phases that can be either miscible, or wholly or partially immiscible, as well as a vapor phase. Despite a great deal of both theoretical and experimental work on this problem by van der Waals and his successors, work which produced much useful knowledge about the various types of phase equilibria that are possible in fluid mixtures, complete solutions to the problem were only obtained after 1967, when the availability of modern computers made calculations of mathematical problems of this complexity feasible for the first time. The results obtained were, in Rowlinson's words,