x = b cos (στ + ε), y = b sin (σt + ε),
(28)
where b, ε are arbitrary, and
| σ = | γa/c | . |
| 1 + M0a2/C |
(29)
The circle is described with the constant angular velocity σ.
When the gravity of the rolling sphere is to be taken into account the preceding method is not in general convenient, unless the whole motion of G is small. As an example of this latter type, suppose that a sphere is placed on the highest point of a fixed sphere and set spinning about the vertical diameter with the angular velocity n; it will appear that under a certain condition the motion of G consequent on a slight disturbance will be oscillatory. If Oz be drawn vertically upwards, then in the beginning of the disturbed motion the quantities x, y, p, q, P, Q will all be small. Hence, omitting terms of the second order, we find
M0ẍ = P, M0ẏ = Q, R = M0g,
Cṗ = −(M0ga/c) y + aQ, Cq̇ = (M0ga/c) x − aP, Cṙ = 0.
(30)
The last equation shows that the component r of the angular velocity retains (to the first order) the constant value n. The geometrical relations reduce to