The subject of this paper is spatial finite-difference approximations to the discrete-ordinate equations for azimuthally symmetric transport of monoenergetic particles in plane-parallel geometry. In previous work it has been shown that almost allclosed linear one-cell functional(CLOF) methods are equivalent to a linear finite-difference method. Here it is shown that any such method that is consistent, in a certain natural sense, is equivalent to a consistent one-step finite-difference approximation to a linear two-point boundary-value problem, in the sense of earlier work of Keller and White. It follows, from observations in this same work, that for CLOF methods consistency impliesbothconvergence and stability.