A method of Atkinson ( Numer. Math. 10 (1967), 117–124) for solving Fredholm equations of the second kind with nonunique solutions is applied to problems of plane strain/stress of a fibre-reinforced composite. Morland ( Int. J. Solids Struct. 9 (1973), 1501–1518) has shown that the problem may be reduced to a potential problem with a nonstandard-boundary condition relating pairs of boundary values at the ends of the fibre chords. Under traction-boundary conditions an integral representation of the displacements leads to an equation with nonunique solutions. A careful examination of the results of Atkinson's method was made and the stress and displacement fields computed for an ellipse under a unidirectional-tensile loading. Plots of the hoop stress are given for a wide range of fibre directions, traction profiles and directions, eccentricities, and material constants.