The equations of electrostatic irrotational space‐charge flow are set up in a paraxial manner. Given an arbitrary trajectory, and the potential distribution along that trajectory, the variation of such quantities as potential and beam convergence in the vicinity of the specified trajectory are investigated. A detailed analysis is carried out for beams from a space‐charge‐limited cathode in the form of a cone. The agreement between the paraxial theory and the exact theory for particular beams is investigated. Numerical results are presented for beams from a conical cathode which predict beams with a microperveance of about 15 and a very large area convergence. It is felt that by use of this method much higher perveances and beam convergences could be obtained for hollow and sheet beams than have heretofore been possible.