Constructive theory of the unicursal cubic by synthetic methods

Abstract

1. SCHROETER'S t classic work on the general cubic leaves little to be desired in point of symmetry and generality. It is nevertheless interesting to build up the theory of the unicursal cubic, the curve being defined as the locus of the intersection of corresponding rays of two projective pencils, one of the first and the other of the second order. This has in fact been done by DRASCH. t The following discussion, based likewise on this definition, is materially simplified by the use of the properties of the point designated in ? 7 by E. Incidentally the investigation brings to light a remarkable one-to-one correspoindence between the points of the plane and the line elements on the cubic. 2. The locus described above has at least one and at inost three points in common with any line in the plane. We assume the truth of this theorem, a proof of which may be found in the eleventh chapter of REYE's Geometrie der Lage. Notations.-Throughout this paper we shall use the following notations: The pencil of the first order will be denoted by s, its center by S, and its rays by a, 6, c, etc. The pencil of the second order (and also the conic enveloped by it) will be denoted by K, and the rays by a, /3, 'y, etc. The cubic itself will be denoted by C. 3. THEOREM.-NO point of the cubic C lies within the conic K. 4. THEOREM.-The cubic C touches the conic K in at least one point, and at most in three. To prove this take S', a point on K, for the center of a pencil s' of the first order perspective to K. This pencil generates with s a conic which cuts KC in at least one and at most three other points besides S'. These are easily seen to be points on C.