where the dots indicate derivatives with respect to the time. The functions 4, *, X are assumned to have partial derivatives of first and second order in the region of space considered. The case where the force vanishes everywhere is excluded. The motion of the particle is determined when its initial position and initial velocity are given. By taking all possible initial conditions we obtain a definite quintuply infinite system of trajectories. Our object is to study the properties of such systems of curves with a view to obtaining a complete geometric characterization. The main result is stated in article 39 at the end of the paper. The first properties derived are consequences of the elementary fact that the osculating plane at any point of a trajectory is determined by its initial direction and by the direction of the force (articles 4-11). The next set relate to osculating, spheres.t With each lineal element there are associated oo1 osculating spheres whose centers lie on a straight line (articles 12-15). The straight lines corresponding to all the elements at a point form a congrulence of order one and class three determined by a twisted cubic curve (articles 16-23). The properties obtained at this stage belong to a more general class of curve systems than the dynamical class. The final characterization is attained by introducing certain related systems of plane curves, here termed S-systems, which have all the