A KINEMATICAL THEOREM

Abstract

TAKE a plane, and, for clearness of idea, consider it as fixed horizontally. On this fixed plane lay another, and throughout the subsequent movement let the surfaces of the two planes always remain in contact. Now let the upper plane, starting from any position, be moved about in any manner whatever, making any number (N) of rotations, the points on it describing curves of any desired degree of complexity on the lower plane; and let it finally settle down again into its initial position, the curves described by the points on it being, in consequence, closed curves. Take the upper plane, and let us investigate the position on it of those points which have described curves of any given area (A) on the fixed plane. However complex the curves described by them may be, the points will be found to form a circle on the upper plane; and if we give to A different values, the corresponding circles will be found to be all concentric Further, if we call the circle corresponding to the value A=0 the zero-circle, the area of the curves described by the points on any other circle of the system equals N times the ring inclosed between that circle and the zero-circle. It is remarkable that such a singular point as the centre of the circles should exist.