A ruled surface of the fourth order with high symmetry and a curve with four cusps

Abstract

As is known, these surfaces of the fourth order can be generated by two projective quadratic primas of planes. The theory of these surfaces is expounded in detailed manuals. In the general case, the surface possesses a hexaprima of points at which two rays of the surface intersect. Planes passing through such a pair of rays can be considered tangent planes, and since the plane section of the surface is in the general case a curve of the fourth order, it is clear that if a plane passes through one of the rays of the surface, this curve decomposes into this ray and a curve of the third order, and if a plane passes through a pair of rays, the same curve decomposes already into this pair and a conoprima; hence we see that tangent planes passing through pairs of rays of the surface intersect this surface further in a conoprima.

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