### Plane Cubics Associated with the Quadrangle-Quadrilateral Configuration

The Annals of Mathematics, 2nd Ser., Vol. 26, No. 1/2 (Sep. - Dec., 1924), 47-58

Introduction

It is know that in the special Desargues configuration 10_{3}, called the quadrangle-quadrilateral configuration, the associated four-point and four-line are each covariant to the other. In a former paper by the writer it was shown that four real points chosen arbitrarily in the plane as critical centers (vertices of inflexional triangles), determine a syzygetic pencil of cubics as one of the four. It appears that this quadruple of pencils of cubics is projectively associated with the quadrangle-quadrilateral configuration. The points and lines of the 10_{3} can be interpreted in terms of the cubics, the points of inflexion and critical centers for the cubics have unique positions on the ten lines, and the four Hessian configurations (9_{4}, 12_{3}) unite into a configuration (18_{7}, 42_{3}).

It further appears that the points and lines of the quadrangle-quadrilateral configuration, together with those of the diagonal triangle common to the associated 4-point and 4-line, can be interpreted in terms of a second quadruple of pencils of cubics also projectively associated with the 10_{3}.

Furthermore, the points of inflexion of the four sets of thirteen pencils of cubics, i.e., of the four configurations C_{13}, of which each of the quadruples is the origin, all lie on the configuration and diagonal lines of the 10_{3}.

Klein's group G_{4!} which carries the 4-point and 4-line into themselves, also leaves invariant each of the quadruple of pencils of cubics, and each of the sets of four configurations C_{13}.