The second-order differential equation governing the non-radial oscillations of polytropic stellar models has been integrated for values of the polytropic index varying between 3 and 4. It has been shown that the fundamental mode is no longer a member of the discrete frequency spectrum for the polytrope n = 3.25; and that, as n increases, an increasing number of modes on either side of the fundamental mode disappear from the spectrum. It has been shown analytically that the equation has no solutions compatible with the boundary conditions of the problem for the limiting polytrope n =5.