Truncation of the Spherical Harmonic Expansion of the Transport Equation

The most general linear truncation relation for the spherical harmonic representation of the transport equation in three dimensions is shown in any order to be a partial differential equation. This equation is uniquely determined up to two independent scalar parameters in the time-dependent case and one scalar parameter in the time-independent case. In the time-dependent situation, one of the parameters may be related to the other parameter, which is pertinent to the time-independent limit, in such a way as to give correct retardation in all orders.