The even-order spherical-harmonics theory for cylindrical geometry is developed along the same lines previously utilized for slab geometry. In particular an ‘effective boundary moment’ is found such that the common spherical-harmonics approach can be straightforwardly applied. The disadvantage-factor problem for a cylindrical unit cell is utilized to show the inherent countervergence of the odd- and even-order results when utilized in this manner. An extrapolation procedure is suggested to overcome the difficulty of divergence for small unit-cell sizes.