Air flows past a fixed thin body of a general planform, at a non-uniform small clearance from a plane ground surface. The flow beneath the body is described by a linear two-dimensional partial differential equation, in which the clearance appears as an input non-constant coefficient. Solutions are required subject to separate leading-edge and (nonlinear) trailing-edge boundary conditions, at the edge contour of the planform. The transition points between leading and trailing edge are not necessarily at the lateral extremities of this contour, and are to be determined as part of the solution. As an illustration, a solution is obtained for a circular planform with an exponentially varying clearance. The general problem is relevant to vehicle aerodynamics, especially for racing cars, and some qualitative discussion of the nature of the negative-lift ground-effect problem for such vehicles, and of the effect of ‘skirts’, is presented here.