  The hypersurface of Luroth quartic curves inside the projective space of
plane quartics has degree 54. We give a proof of this fact along the lines
outlined in a paper by Morley, published in 1919. Another proof has been given
by Le Potier and Tikhomirov in 2001, in the setting of moduli spaces of vector
bundles on the projective plane. Morley's proof uses the description of plane
quartics as branch curves of Geiser involutions and gives new geometrical
interpretations of the 36 planes associated to the Cremona hexahedral
representations of a nonsingular cubic surface.
