The electrostatic potential problem involving a dielectric elliptic cylinder and an external parallel line charge is solved in terms of continuous image charge and dipole layer distributions on the focal strip of the cylinder. As a starting point, two integral identities are derived from the relation between the true charge distributions on a flat conducting strip induced by a parallel line charge and the corresponding series expansion of the potential. It turns out that two different solutions must be applied, depending on the distance of the source and shape of the cylinder. By letting the cross-section of the cylinder approach a circle, the image is seen to coincide with the line image solution available for the circular cylinder. A simple expression is given for the equivalent polarisation distribution on the focal strip due to a homogeneous field.