A plane quartic curve with twelve undulations

where x, y, z are homogeneous coordinates in a plane, was encountered by Ciani [Palermo Rendiconli, Vol. 13, 1899] in his search for plane quartic curves that were invariant under harmonic inversions. If x, y, z undergo any permutation the ternary quartic form on the left of (1) is not altered; nor is it altered if any, or all, of x, y, z be multiplied by — 1. There thus arises an octahedral group 0 of ternary collineations for which every curve of the pencil is invariant. Since (1) may also be written