Analytical and numerical solutions of similarity form are obtained for (assumed) inviscid flow in the generating plane of an axi-symmetric conical hydrocyclone below the level of the vortex finder. Results are compared with those based on corresponding approximate solutions derived using the methods of Bloor and Ingham, and differences from the original Bloor and Ingham solution are highlighted. Solutions are found which, if realised physically, would lead to greater separation efficiency of particles with density greater than the fluid. Such solutions are characterised by vertical velocity profiles at a given height which are flatter, and inward radial velocities which are smaller, in a region away from the cyclone wall and the axis of symmetry. However, entry conditions under which these solutions can occur are not known.