A complete theory of the non-radial oscillations of a static spherically symmetric distribution of matter, described in terms of an energy density and an isotropic pressure, is developed, ab initio, on the premise that the oscillations are excited by incident gravitational waves. The equations, as formulated, enable the decoupling of the equations governing the perturbations in the metric of the space-time from the equations governing the hydrodynamical variables. This decoupling of the equations reduces the problem of determining the complex characteristic frequencies of the quasi-normal modes of the non-radial oscillations to a problem in the scattering of incident gravitational waves by the curvature of the space-time and the matter content of the source acting as a potential. The present paper is restricted (for the sake of simplicity) to the case when the underlying equation of state is barotropic. The algorism developed for the determination of the quasi-normal modes is directly confirmed by comparison with an independent evaluation by the extant alternative algorism. Both polar and axial perturbations are considered. Dipole oscillations (which do not emit gravitational waves), are also treated as a particularly simple special case. Thus, all aspects of the theory of the non-radial oscillations of stars find a unified treatment in the present approach. The reduction achieved in this paper, besides providing a fresh understanding of known physical problems when formulated in the spirit of general relativity, provides also a basis for an understanding, at a deeper level, of Newtonian theory itself.