On the focal surfaces of ray systems and the singularity surfaces of complexes

Any line is associated with a two-parameter group of polar complexes relative to a given first-order line complex that are called tangential complexes when the line belongs to the complex. The congruence that is determined by the tangential complex of a complex line is of a special kind, in that both of its directrices coincide in a single line that is the complex line itself. It is of a special kind of line that Plücker referred to as the singular lines of the given complex, for which the tangential complexes are all special, and the directrices of the congruence therefore define a pencil of rays in a plane that goes through the singular lines, namely, the associated singular plane, that has a midpoint that lies on the singular line, namely, the associated singular point. Plücker examined the distinguished elements thus defined more closely while restricting himself to seconddegree complexes. For them, he arrived at the conclusion that the associated singular point is the fixed contact point of the singular line with all of the complex curves that it contacts whose associated singular planes are the common tangential planes to the complex cone that includes all of the singular lines. He further recognized the identity of the singular lines with the lines of intersection of those pairs of planes that define a complex cone, and with the connecting lines of those point-pairs that would represent a complex curve. The vertex of the complex cone is, in any event, the associated singular point, and the plane of the complex curve is the associated singular line. In particular, Plücker examined the fourth-order surfaces that are defined by the singular points of second-degree complexes and the fourth-order surface that is enveloped by its singular planes, and found that these two surfaces are identical (), in such a way that at any point of the surface, any tangent is a singular line that corresponds to the point as a singular point and the tangent plane as a singular plane. The validity of the latter theorem of Plücker for complexes of arbitrary degree is proved in my Habilitationsschrift () by simple geometric considerations that are based upon the double definition of the focal surface of a ray system. In fact, the surface of singular points or planes defines a part of the focal surface of the congruence that is composed of the singular lines. The proof of the identity of both definitions of the singularity surfaces of a complex − or more generally, the focal surface of a ray system –