In the usual Berry (I) expression of the generalised theorem of three moments (i.e., generalised in the sense that end load deflection effects are taken into account) the lateral loading on a continuous beam is regarded as uniformly distributed over each bay into which the beam length may be divided (either naturally by supports or artificially for convenience of analysis). In this paper the lateral loading is taken as distributed parabolically so that the resulting equations may be used as a sort of Simpson's rule for dealing with any arbitrary loading. By taking into account the “ sinking “ of supports any particular beam can be divided up into a convenient number of bays over each of which the lateral loading may be assumed to be parabolic. We may thus obtain reasonably accurate solutions for the bending moments, etc., in a continuous beam under almost any type of lateral loading.