Dynamical Trajectories in a Resisting Medium

This is not an arbitrary differential equation of the third order. We have given a completely characteristic set of five geometric properties of the dynamical trajectories in the Princeton Colloquium. The dynamical trajectories are projectively invariant, and recently Terracini has transformed each of the five properties into purely projective language.3 2. In the subsequent discussion, we shall need only three of the characteristic sets of five geometric properties. These may be stated in the following manner. Property I. If for each of the co trajectories passing through a given point in a given direction we construct the osculating parabola at the given point, the locus of the foci of these parabolas is a circle passing through that point. The most general systems of X I curves possessing the Property I are given by differential equations of the forms4