The spherical harmonic (P3) approximation to the Boltzmann equation is applied to the case of a finite cylinder, with symmetry about the axis of the cylinder. Solutions are obtained for the case of a neutron source proportional to cos Bzz where z is measured along the axis of the cylinder and Bz2 is the axial buckling. These solutions are then expanded in terms of Bz and only terms of order Bz2 or less are retained. The approximate solutions are then used to calculate the thermal utilization of a cell of finite height composed of a natural uranium rod surrounded by a D2O moderator as a function of the axial buckling. The resultant expression for the utilization has the form where f(0) is the utilization of the cell of infinite height and the constant L2 corresponds to the thermal diffusion area in two-group theory. Results are obtained for several cells and compared with those obtained using other calculational methods.