I = M (α2λ2 + β2μ2 + γ2ν2) = Mp2,
(42)
if p denotes the perpendicular drawn from O in the direction (λ, μ, ν) to a tangent plane of the ellipsoid
| x2 | + | y2 | + | z2 | = 1 |
| α2 | β2 | γ2 |
(43)
This is called the ellipsoid of gyration at O; it was introduced into the theory by J. MacCullagh. The ellipsoids (41) and (43) are reciprocal polars with respect to a sphere having O as centre.
If A = B = C, the momental ellipsoid becomes a sphere; all axes through O are then principal axes, and the moment of inertia is the same for each. The mass-system is then said to possess kinetic symmetry about O.
If all the masses lie in a plane (z = 0) we have, in the notation of (25), c2 = 0, and therefore A = Mb2, B = Ma2, C = M(a2 + b2), so that the equation of the momental ellipsoid takes the form
b2x2 + a2y2 + (a2 + b2) z2 = ε4.
(44)