The formal object of this note is the calculation of the principal moments of inertia of a set of particles at their mass centre, in terms of their mutual distances; he calculation brings in some identities which although simple may be in part novel.

A considerable amount is known about the latent roots of matrices of the form

in the case when each cross-product of non-diagonal elements, aici-1, is positive. One forms the sequence of polynomials fr(λ) = |Lr−λI| for r = 1, 2, … n, and observes that

then it is easy to deduce that (i) the zeros of fn(λ) and fn_1(λ) interlace—that is, between two consecutive zeros of either polynomial lies precisely one zero of the other (ii) at the zeros of fn(λ) the values of fn-x(λ) are alternately positive and negative, (iii) all the zeros of fn(λ)— i.e. all the latent roots of Ln—are real and different.

It seems a pity that Hamiltonian dynamics—contact transformations and so on—is regarded as a fearsome subject, too time-consuming to teach to most students; for it is the one branch of dynamics to point a way to new developments in this century. Moreover the basic ideas are extremely simple, but presented in an unfortunate way in all the text-books.

A dynamical system means here a system specified by generalised coordinates qα(α = 1, 2, …, n) and a Lagrangian L which is a quadratic polynomial in the generalised velocities, say

(with a summation convention).