Agnes Scott College

B. M. Turner

Plane Cubics with a Given Quadrangle of Inflexions
American Journal of Mathematics, Vol. 44, No. 4 (Oct. 1922), 261-278


That every non-singular cubic has nine points of inflexion, lying in related positions on the curve, is a classical fact of mathematics. Of these points four may be chosen arbitrarily; and when such a quadrangle is fixed, the finding of the position of the remaining five presents a question worthy of consideration. It appears that all the sets of five combine into a group of fifteen points whose relative positions with respect to the given four depend upon equianharmonic properties; but that the equianharmonic relations follow as a consequence of a combination of harmonic relations and hence, in a number of cases, the points may be determined by linear and quadratic constructions.*

It is also well known that four points of inflexion, no three collinear, impose eight conditions on a cubic and determine it as one of a singly infinite system; but, since only four of the conditions are linear while the other four are of the third degree, the system is not a pencil. It will be shown that the four points determine a system consisting of six pencils and that every two of the six have a fifth point of inflexion in common, that is, through every one of the fifteen points two of the pencils pass, and have consequently an inflexion.

*That the whole set of nine points depends only on quadratic constructions was virtually shown by Möbius in the determination of two quadrangles in- and circumscribed to one another. The eight vertices with the addition of the one common diagonal point form the desired set.