Again, the Periodic Time in a circle whose radius is 540. Let the distance of the vertex of the parabola from its focus be a, then the velocity in the curve is Again, if ẞ be the velocity with which the system moves, and y' the ordinate of the curve-in-space at the end of the time t, the whole increment of y' during dt is dy Bdt dy according as the velocity to the axis arising from the motion in the curve, is in the same or opposite direction to that of the system. Hence, putting a + 1 = A, and ‚√ (1 + a + — x + x = X a a 1. (2x + A + 541. radius = r, is x2) The velocity in a circle, when the force is F, and where is the absolute force, and or that same function of the radius () according to which the central force attracts at different distances. Hence, at the same distance, and for the same law of force denoted by 4, we have the law required. 543. periodic time. Let a, b be the given semi-axes, and T the given Then since the force tends to the focus, if and be the polar co-ordinates, it is easily shewn (see Newton, Prop. 1.) that the areas described by ę are proportional to the times of deв scription; and because the area of the ellipse is ab, we have T: rab:: dt: 2 ds e2 de 2rab 2rab ds T which gives the velocity at any point of the ellipse. c2 dp F= |