the equation (31) becomes

x2+ y2+ z2= 1;
α2 + θ β2 + θγ2 + θ

(33)

for different values of θ this represents a system of quadrics confocal with the ellipsoid

x2+ y2+ z2= 1,
α2 β2γ2

(34)

which we shall meet with presently as the “ellipsoid of gyration” at G. Now consider the tangent plane ω at any point P of a confocal, the tangent plane ω′ at an adjacent point N′, and a plane ω″ through P parallel to ω′. The distance between the planes ω′ and ω″ will be of the second order of small quantities, and the quadratic moments with respect to ω′ and ω″ will therefore be equal, to the first order. Since the quadratic moments with respect to ω and ω′ are equal, it follows that ω is a plane of stationary quadratic moment at P, and therefore a principal plane of inertia at P. In other words, the principal axes of inertia at P arc the normals to the three confocals of the system (33) which pass through P. Moreover if x, y, z be the co-ordinates of P, (33) is an equation to find the corresponding values of θ; and if θ1, θ2, θ3 be the roots we find

θ1 + θ2 + θ3 = r2 − α2 − β2 − γ2,

(35)

where r2 = x2 + y2 + z2. The squares of the radii of gyration about the principal axes at P may be denoted by k22 + k32, k32 + k12, k12 + k22; hence by (32) and (35) they are r2 −θ1, r2 − θ2, r2 − θ3, respectively.