(15) φ + ψ cos θ = R, a constant, will lead to an equation for dθ/dt, or dz/dt, in terms of cos θ or z, the integral of which is of hyperelliptic character, except when A = C′.
In the suspension of fig. 8, the motion given by φ is suppressed in the stalk, and for the fly-wheel φ gives the rubbing angular velocity of the wheel on the stalk; the equations are now
T = ½A (θ2 + sin2 θψ2) + ½C′ cos2 θψ2 + ½CR2 = H + gMh cos θ,
(16)
A sin2 θψ + C′ cos2 θψ + CR cos θ = G,
(17)
and the motion is again of hyperelliptic character, except when A = C′, or C′ = 0. To realize a motion given completely by the elliptic function, the suspension of the stalk must be made by a smooth ball and socket, or else a Hooke universal joint.
Finally, there is the case of the general motion of a top with a spherical rounded point on a smooth plane, in which the centre of gravity may be supposed to rise and fall in a vertical line. Here
T = ½ (A + Mh2 sin2 θ) θ2 + ½A sin2 θψ2 + ½CR2 = H − gMh cos θ,
(18)