481. Generally for conical revolutions, if the length of the string be l, and r the radius of the base, then the body being acted + upon by three forces; (w) in the direction of gravity; another (S) in the direction of the string or its tension; and the third (F) towards the centre of the base of the cone, and therefore centripetal, we have, Again, by 440, the time of revolution in the circular base is From (c) it appears that the times of revolution, for different conical pendulums are always the same for the same altitude which expresses the required time in seconds, since is the 2 space described in a vacuum by a falling body during that interval. 482. Generally let the perihelion distance of the comet ber, and the mean distance of the earth be a. Then, since the mean velocity of the earth is that by which a circle would be uniformly described in the time of the earth's revolution in its orbit, or in the time Σπαξ and p = √Te .. at the extremity of the latus-rectum, where e=2r 483. This very ingenious theorem is easily reduced to the following; If SAC (Fig. 97.) be a parabolic area, S being the focus, and A the vertex of the parabola, and perpendiculars bisecting AS and SC meet in H; then GH is proportional to the area ASC. For the figure being completed, we have HG+GR: RS+SM :: SC: CN C)-GR ᏀᎡ. there being no correction, if t is 0 when pr. But c√(2rp) by (453). = ..the general expression for the describing any arc of a parabola from the perihelion is If we use the expression (a) for the ellipse, whose equation is .(c) and we get for the time between the vertex of the ellipse and the extremity of its latus rectum, T= αν N at . (sin.-1) − e) + — )— √√2. ae √ (1—e2) Again, let ea, then the time from the perihelion to the extremity of the minor-axis is Hence, and from expression (d) it follows that if e or the ratio of the eccentricity to the major axis is given, then the Times from the vertex either to the extremity of the latus-rectum, or to that of the axis-minor, are in the sesquiplicate ratio of the major-axes, and the subduplicate ratio of the absolute forces. |