provided the summation ΣΣ on the left hand be understood to include each pair of particles once only. This theorem, also due to Lagrange, enables us to express the mean square of the distances of the particles from the centre of mass in terms of the masses and mutual distances. For instance, considering four equal particles at the vertices of a regular tetrahedron, we can infer that the radius R of the circumscribing sphere is given by R2 = 3⁄8a2, if a be the length of an edge.
Another type of quadratic moment is supplied by the deviation-moments, or products of inertia of a distribution of matter. Thus the sum Σ(m·yz) is called the “product of inertia” with respect to the planes y = 0, z = 0. This may be expressed In terms of the product of inertia with respect to parallel planes through G by means of the formula (14); viz.:—
Σ (m · yz) = Σ (m · ηζ) + Σ (m) · yz
(21)
The quadratic moments with respect to different planes through a fixed point O are related to one another as follows. The moment with respect to the plane
λx + μy + νz = 0,
(22)
where λ, μ, ν are direction-cosines, is
Σ {m (λx + μy + νz)2} = Σ (mx2)·λ2 + Σ (my2)·μ2 + Σ (mz2)·ν2 + 2Σ (myz)·μν + 2Σ (mzx)·νλ + 2Σ (mxy)·λμ,