The inviscid aerodynamics of high-aspect-ratio wings having conical surfaces that are inextensible membranes capable of carrying compressive stresses and supported along two straight generators are investigated. For a given shape, the aerodynamics of the wing are found using lifting-line theory and thin-airfoil theory. The observation that the line of action of the membrane tension along each supporting generator is the same gives rise to a global constraint between the membrane shape, the pitching moment, and the root bending moment. This constraint enables an approximate solution to be found for the equilibrium membrane shape without needing to consider the full equations of equilibrium for the membrane. The functional form of the relationships between the wing shape and its force and moment coefficients is found to be analogous to those for the corresponding two-dimensional problem. The wing shapes considered in detail are the wing shape needed for elliptic loading, a triangular planform and an optimum wing at off-design conditions