(19)
The last equation shows that the angular velocity about the normal to the plane is constant. Again, since the point of the sphere which is in contact with the plane is instantaneously at rest, we have the geometrical relations
u + qa = 0, v + pa = 0, w = 0,
(20)
by (12). Eliminating p, q, we get
(M0 + Ca−2) u̇ = X, (M0 + Ca−2) v̇ = Y.
(21)
The acceleration of the centre is therefore the same as if the plane were smooth and the mass of the sphere were increased by C/α2. Thus the centre of a sphere rolling under gravity on a plane of inclination a describes a parabola with an acceleration
g sin α/(1 + C/Ma2)
parallel to the lines of greatest slope.