The special case where both cones are right circular and ω is constant is important in astronomy and also in mechanism (theory of bevel wheels). The “precession of the equinoxes” is due to the fact that the earth performs a motion of this kind about its centre, and the whole class of such motions has therefore been termed precessional. In fig. 78, which shows the various cases, OZ is the axis of the fixed and OC that of the rolling cone, and J is the point of contact of the polhode and herpolhode, which are of course both circles. If αbe the semi-angle of the rolling cone, β the constant inclination of OC to OZ, and ψ̇ the angular velocity with which the plane ZOC revolves about OZ, then, considering the velocity of a point in OC at unit distance from O, we have
ω sin α = ±ψ̇ sin β,
(3)
where the lower sign belongs to the third case. The earth’s precessional motion is of this latter type, the angles being α = .0087″, β = 23° 28′.
If m be the mass of a particle at P, and PN the perpendicular to the instantaneous axis, the kinetic energy T is given by
2T = Σ {m (ω·PN)2 } = ω2·Σ (m·PN2) = Iω2,
(4)
where I is the moment of inertia about the instantaneous axis. With the same notation for moments and products of inertia as in § 11 (38), we have
I = Al2 + Bm2 + Cn2 − 2Fmn − 2Gnl − 2Hlm,
and therefore by (1),