A perturbation method is derived forr-modes in a slowly and uniformly rotating star. In contrast to previous studies, the perturbation of the gravitational potential is included in the perturbation method. On the assumption that the effects of the centrifugal force are taken into account in the equilibrium model up to the second order in the angular velocity, an eigenvalue problem of sixth-order in the radial coordinate is derived that allows one to determine the zeroth-order toroidal displacement field and the third-order term in the expansion of the eigenfrequency. Furthermore, another eigenvalue problem is derived that governs the first-order toroidal displacement field and the fourth-order term in the expansion of the eigenfrequency. This second eigenvalue problem is also of the sixth-order in the radial coordinate. It is shown that the third-order term in the expansion of the eigenfrequency is real, and that the fourth-order term is zero.