Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/115

 every direction it trikes the circle, will be as its velocity: and therefore the um of the forces, in a given time, will be as that velocity and the number of reflexions conjunctly; that is, (if the pecies of the polygon be given) as the length decribed in that given time, and increaed or diminihed in the ratio of the ame length to the radius of the circle; that is, as the quare of that length applied to the ratios: and therefore if the polygon, by having its ides diminihed is increead, coincides with the circle, as the quare of the arc decribed in a given time applied to the radius. This is the centrifugal force, with which the body impells the circle; and to which the contrary force, wherewith the circle continually repells the body towards the centre, is equal.

There being given in any places, the velocity with which a body deribes a given figure, by means of forces directed to ome common centre; to find that centre. Pl. 3. Fig. 1.

Let the three right lines PT, TQV, VR touch the figure decribed in as many points P, Q, R, and meet in T and V. On the tangents erects the perpendiculars PA, QB, RC, reciprocally proportional to the velocities of the body in the points P, Q, R, from which the perpendiculars were raied, that is, o that PA may be to QB as the velocity in Q to the velocity in P, and QB to RC as the velocity in R to the velocity in Q; Thro' the ends A, B, C, of the perpendiculars draw AD, DBE, EC, at right angles,