be the axis of the cycloid, then the velocity gx2 Hence the time of flight is = Again, the time of an oscillation is 425. 21 Since, generally, the time of an oscillation is 426. Let be the length of the pendulum; then 21 = length of the whole cycloid, and, since the time of an oscillation is 427. t, = 2 t:t,::: 2. Let the body descend from rest through the arc FA (Fig. 90); then the greatest velocity will be at the lowest point A ; and, since s cc = v2, if BR = BA, the velocity of a body de 1 4 scending from rest through BA at R will = velocity at A. 2 Hence, through R drawing RM LAB, and meeting the curve in M, the point M is determined in which the velocity of the body 1 descending down the cycloid = the greatest velocity. 428. Since generally, v2 = 2gs, the velocity due to 2r (r being the radius of the generating 2 of the cycloid), is Now the velocity V of the moving point in the cycloid: velocity of the moving point in the circle :: ds: dz, s and z denoting the cycloidal and circular arcs respectively. x being the abscissa corresponding to s, measured from the vertex along the axis of the cycloid. a body oscillating in the cycloid. (Venturoli, p. 102.) 429, In the generating O AB (Fig. 90,) inflect AP its radius, and draw R'PM'AB; then since AM' = 2AP = AB = AF, AF is bisected in M', Now (Translat. of Venturoli, p. 103.) the time through any arc of the cycloid, whose abscissa, measured from the lowest point is x, is Then t= Co 2g - 1) = √ / × 120° But when x = 0, the time (t) down FA is 2g 430. have t: 2t, 1: 3. In Newton's Construction (Prop. 50, Princep.), we CA: CO: CO: CR or AO+CO: CO :: CO : CO - OR .. AO CO: OR: CO- OR. But in the common cycloid, SOQ is a straight line, and .. CO 88. .. COCO - OR, and AO Also BW is parallel to RA, and BV .. PT = 2PV = PS, &c. &c. OR; AO OR = VW 431. Let R be the radius of the base, r that of the wheel; then (by Prop. 49, Princip. Newt.) it easily appears that Hence s T 1000 2, the space required. 433. Let be the length of either pendulum; then the distance descended vertically from the highest to the lowest points of the and cycloid, will be 2 I and 2 and the velocities in the curves (v, v') will be those due to such distances. Now the chord (C) of the quadrant 2 the arc (A) of 434. The semi-cycloidal arc FA (Fig. 90) is bisected by |