In I we obtained an equation, called the "chromatic equation", for the generating function g(x, y, z, λ). In II and III we obtained special equations, valid in the cases λ = τ + 1 and λ = 3 respectively, for the generating function l(y, z, λ), defined as the coefficient of x2 in g(x, y, z, λ). The argument was independent of that in I and no attempt was made to derive the new formulae from the chromatic equation.