A possible electrostatic lens consists of a cylindrical surface along which the potential varies as the square of the distance from some original plane. If the cylinder is very long or is terminated without distorting the field within, the trajectories of charged particles inside it can be calculated analytically. These calculations show that a small diameter source of electrons inside the cylinder can be imaged, with magnification unity and slight aberration, at another point in the cylinder. A short section of the cylinder with open ends serves as a thick lens for focusing a beam of particles, the focusing effect being due to the main field inside the cylinder plus end effects. Such a lens was constructed, the boundary condition being realized by a succession of closely spaced rings maintained at the proper potentials. This lens was more efficient in focusing a highly divergent beam of ions than were several combinations of conventional lenses which were also tried.