which is parallel to (22). It appears from (24) that through any assigned point O three rectangular axes can be drawn such that the product of inertia with respect to each pair of co-ordinate planes vanishes; these are called the principal axes of inertia at O. The ellipsoid (26) was first employed by J. Binet (1811), and may be called “Binet’s Ellipsoid” for the point O. Evidently the quadratic moment for a variable plane through O will have a “stationary” value when, and only when, the plane coincides with a principal plane of (26). It may further be shown that if Binet’s ellipsoid be referred to any system of conjugate diameters as co-ordinate axes, its equation will be

x′2+ y′2+ z′2= 1,
a′2 b′2c′2

(27)

provided

Σ(mx′2) = Ma′2,   Σ(my′2) Mb′2,   Σ(mz′2) = Mc′2;

also that

Σ(my′z′) = 0,   Σ(mz′x′) = 0,   Σ(mx′y′) = 0.

(28)

Let us now take as co-ordinate axes the principal axes of inertia at the mass-centre G. If a, b, c be the semi-axes of the Binet’s ellipsoid of G, the quadratic moment with respect to the plane λx + μy + νz = 0 will be M(a2λ2 + b2μ2 + c2ν2), and that with respect to a parallel plane

λx + μy + νz = p