The Lagrangian and Hamiltonian expressions for the kinetic energy of a system of $N$ particles are obtained in such a form that the rotational, vibrational and coupling terms may be distinguished. The principal axes of inertia are used to define rotation. The ordinary moments of inertia appear in the Lagrangian kinetic energy but these are replaced by other functions of the radii of gyration in the Hamiltonian. This throws doubt upon all molecular configurations assigned on the basis of empirical values of the moments of inertia.