Of the properties of a tangent it may be noticed that the tangent at any point is equally inclined to the focal distances of that point; that the feet of the perpendiculars from the foci on any tangent always lie on the auxiliary circle, and the product of these perpendiculars is constant, and equal to the product of the distances of a focus from the two vertices.
But, p and q being respectively the perpendiculars to the lines of action of the forces, this equation reduces to Pp=Rq, FIG.
Hence also the ratio of the com ponents of the velocities of two points A and B in the directions AP and BW respectively, both in the plane of rotation, is equal to the ratio of the perpendiculars Fni and Fn.
Application to a Pair of TurnIng Fseces.Let ai, a2 be the angular velocities of a pair of turning pieces; Of, Oi the angles which their line of connection makes with their respective planes of rotation; Ti, r2 the common perpendiculars let fall from the line of connection upon the respective axes of rotation of the pieces.
Then the equal components, along the line of connection, of the velocities of the points where those perpendiculars meet that line are airi cos 0i = afri cos Oi; consequently, the comparative motion of the pieces is given by the equation ai_rieos0i ~I