A modified truncation of the P1 equations for the treatment of multidimensional time-dependent neutron transport is presented that avoids some inconvenient features of the usual PL· approximation, such as the nonuniqueness of the stationary equations in vacuum and the discontinuity of certain moments at material interfaces. The mathematical properties of the original (PL) and modified (EPL) approximations, together with interface and vacuum boundary conditions, are compared. An approximate solution method for both types of equations is derived from a variational principle, and numerical results are given for time-dependent P1 and EP1 calculations in two-dimensional cylindrical geometry.