A Variable System of Sevens on Two Twisted Cubic Curves.

Seven points chosen at random on a twisted cubic curve, like six points on a conic in the plane, give rise to a distinctive theorem; for as five points determine a conic, so the twisted cubic is determined by six points. In the case of the conic, this is the theorem of the Pascal hexagon, six points in a definite order leading to a definite line. Conic and line remaining fixed, the hexagon may vary with four degrees of freedom. In the case of the twisted cubic, not a mere sequence of the seven points, but an arrangement of them in seven triads, determines seven planes, and the theorem states that these planes are all osculated by a second twisted cubic curve. So much was established by a direct proof in these PROCEEDINGS in August, 1915; but the question of variability, whether the points and planes are free to move while the two curves remain fixed, was not examined. Now it is found that the system is variable with one degree of freedom. Full proof is contained in a paper soon to appear in the Transactions of the American Mathematical Society. The following is an outline. Every twisted cubic C3 is a rational curve, and the homogeneous coordinates of its points are cubic functions of a variable parameter: