1. INTRODUCTION The liquid drop model (Fierz 1943;Bohr 1952;Jekeli 1952) of the nucleus involves the assumption that the nucleus has a well-defined surface and a uniform charge and mass density. Experiments on the elastic scattering of high energy electrons (Hofstadter, Fechter, and McIntyre 1953;Hofstadter et al. 1954) have demonstrated that these assumptions are not valid even for heavy nuclei. An attempt is made to obtain a modified hydro dynamical model which allows for non-uniform nuclear charge and mass density distributions. In the liquid drop model distortions of the surface of the nucleus are considered; in this model similar distortions are considered not of the surface but of the shape of the nucleus, the distortions having effect throughout the whole nucleus. Excited states of the nucleus are then due to oscillations of the shape of the nucleus, i.e. oscillations involving: departures from spherical symmetry of the mass and charge distributions of the nucleus.Longitudinal compressional waves may be expected in the nucleus, involving nodes in the radial density distributions. As in the liquid drop model (Bohr 1952), the energy of excitation of these modes of radial oscillation should be' much greater than the energy of excitation of the shape oscillations. Thus, in treating low-lying nuclear energy levels, it is assumed that the radial dependence of the density distributions does not alter appreciably. This shape oscillation model is then used to consider y-transitions and electron excitation of nuclei. For a uniform charge distribution this model reduces to the usual liquid drop model.