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The equation of the locus in homogeneous rectangular coordinates, x, y, z, is developed under the assumptions: i. The seven cusps are distributed at equal intervals about the unit-circle, their cuspidal tangents concurrent at (0, 0, 1). ii. The curve is invariant under rotations about the origin through angles 2irk/l, (k = 1, 2, 3, • • • , 6). iii. Each cuspidal tangent is an axis of symmetry. As an immediate consequence of ii and iii, one node and one point of inflexion must lie on each cuspidal tangent. The most general equation of a locus of the seventh order is

A self-dual septimic possessing seven of each kind of the simple singularities, and autopolar by seven rectangular hyperbolas and a circle