If the co-ordinate axes be taken to coincide with the principal axes of inertia at O, at the instant under consideration, we have the simpler formulae
2T = Ap2 + Bq2 + Cr2,
(8)
λ = Ap, μ = Bq, ν = Cr.
(9)
It is to be carefully noticed that the axis of resultant angular momentum about O does not in general coincide with the instantaneous axis of rotation. The relation between these axes may be expressed by means of the momental ellipsoid at O. The equation of the latter, referred to its principal axes, being as in § 11 (41), the co-ordinates of the point J where it is met by the instantaneous axis are proportional to p, q, r, and the direction-cosines of the normal at J are therefore proportional to Ap, Bq, Cr, or λ, μ, ν. The axis of resultant angular momentum is therefore normal to the tangent plane at J, and does not coincide with OJ unless the latter be a principal axis. Again, if Γ be the resultant angular momentum, so that
λ2 + μ2 + ν2 = Γ2,
(10)
the length of the perpendicular OH on the tangent plane at J is
| OH = | Ap | · | p | ρ + | Bq | · | q | ρ + | Cr | · | r | ρ = | 2T | · | ρ | , |
| Γ | ω | Γ | ω | Γ | ω | Γ | ω |