  Newton's Proposition 6 generalizes this result to not necessarily uniform motion under centripetal forces along an arbitrary trajectory in which equal areas are swept out in equal times with respect to S, in accord with Proposition 1 of Book 1. The central force at P is again proportional to the displacement from the tangent QR over a short increment of time divided by the square of this time; but now the time is proportional not to the arc PQ, but to the area swept out, which in the limit as Q approaches P, is the triangular area SPxQT/2. Therefore, to keep a body moving along a given non-circular trajectory, the centripetal force must vary along the trajectory as (1/SP2) — that is, 1/r2 — times the limit of (QR/QT2) as Q approaches P. In the second edition Newton adds a corollary that displays another way of seeing the result as a generalization of uniform circular motion: the centripetal force along the trajectory must everywhere vary as v2/(ρsinSPR), where ρ is the radius of curvature of the trajectory at P. With this, the body can be viewed as driven from one instantaneous circle to the next by the component of force tangential to the motion, a component that disappears in the case of uniform circular motion.