Ax2 + By2 + Cz2 − 2Fyz − 2Gzx − 2Hxy = Mε4
(39)
where ε is an arbitrary linear magnitude, the intercept r which it makes on a radius drawn in the direction λ, μ, ν is found by putting x, y, z = λr, μr, νr. Hence, by comparison with (37),
I = Mε4 / r2.
(40)
The moment of inertia about any radius of the quadric (39) therefore varies inversely as the square of the length of this radius. When referred to its principal axes, the equation of the quadric takes the form
Ax2 + By2 + Cz2 = Mε4.
(41)
The directions of these axes are determined by the property (24), and therefore coincide with those of the principal axes of inertia at O, as already defined in connexion with the theory of plane quadratic moments. The new A, B, C are called the principal moments of inertia at O. Since they are essentially positive the quadric is an ellipsoid; it is called the momental ellipsoid at O. Since, by (12), B + C > A, &c., the sum of the two lesser principal moments must exceed the greatest principal moment. A limitation is thus imposed on the possible forms of the momental ellipsoid; e.g. in the case of symmetry about an axis it appears that the ratio of the polar to the equatorial diameter of the ellipsoid cannot be less than 1/√2.
If we write A = Mα2, B = Mβ2, C = Mγ2, the formula (37), when referred to the principal axes at O, becomes