The kinetic energy T of the motion relative to O will be constant. Now T = 1⁄2Iω2, where ω is the angular velocity and I is the moment of inertia about the instantaneous axis. If ρ be the radius-vector OJ of the momental ellipsoid
Ax2 + By2 + Cz2 = Mε4
(1)
drawn in the direction of the instantaneous axis, we have I = Mε4/ρ2 (§ 11); hence ω varies as ρ. The locus of J may therefore be taken as the “polhode” (§ 18). Again, the vector which represents the angular momentum with respect to O will be constant in every respect. We have seen (§ 18) that this vector coincides in direction with the perpendicular OH to the tangent plane of the momental ellipsoid at J; also that
| OH = | 2T | · | ρ | , |
| Γ | ω |
(2)
where Γ is the resultant angular momentum about O. Since ω varies as ρ, it follows that OH is constant, and the tangent plane at J is therefore fixed in space. The motion of the body relative to O is therefore completely represented if we imagine the momental ellipsoid at O to roll without sliding on a plane fixed in space, with an angular velocity proportional at each instant to the radius-vector of the point of contact. The fixed plane is parallel to the invariable plane at O, and the line OH is called the invariable line. The trace of the point of contact J on the fixed plane is the “herpolhode.”
If p, q, r be the component angular velocities about the principal axes at O, we have
(A2p2 + B2q2 + C2r2) / Γ2 = (Ap2 + Bq2 + Cr2) / 2T,
(3)