Approximate vacuum boundary conditions for a PN approximation are obtained by variational methods. Two stationary principles are proposed, one having what we shall call “odd” Marshak conditions as its natural boundary conditions, and the other having “even” Marshak conditions as its natural boundary conditions. The principles are valid for arbitrary geometry. The odd Marshak conditions are seen to be suitable for an odd-order PN approximation and the even Marshak conditions for an even-order PN approximation. The odd Marshak conditions are precisely the conditions obtained by Vladimirov from an extremum principle in which certain restrictions are imposed on the source and scattering. The present treatment contains no such restrictions.