Take next the case of a sphere rolling on a fixed spherical surface. Let a be the radius of the rolling sphere, c that of the spherical surface which is the locus of its centre, and let x, y, z be the co-ordinates of this centre relative to axes through O, the centre of the fixed sphere. If the only extraneous forces are the reactions (P, Q, R) at the point of contact, we have
M0ẍ = P, M0ÿ = Q, M0z̈ = R,
| Cṗ = − | a | (yR − zQ), Cq̇ = − | a | (zP − xR), Cṙ = − | a | (xQ − yP), |
| c | c | c |
(22)
the standard case being that where the rolling sphere is outside the fixed surface. The opposite case is obtained by reversing the sign of a. We have also the geometrical relations
ẋ = (a/c) (qz − ry), ẏ = (a/c) (rx − pz), ż = (a/c) (py − gx),
(23)
If we eliminate P, Q, R from (22), the resulting equations are integrable with respect to t; thus
| p = − | M0a | (yż − zẏ) + α, q = − | M0a | (zẋ − xż) + β, r = − | M0a | (xẏ − yẋ) + γ, |
| Cc | Cc | Cc |
(24)