The variational method as applied to the monoenergetic integro-differential Boltzmann equation is investigated. It is shown that rendering the Lagrangian stationary with respect to small changes in the directional flux and adjoint directional flux is equivalent to solving the Boltzmann and adjoint Boltzmann equations. Topics discussed include the use of variational weight functions, the inclusion of boundary terms in the functional, the interpretation of a variational optimum for a nonself-adjoint operator, and the second variation. It is shown that, for the general trial function ensemble and within a special restricted trial function ensemble, the variational method is a saddle point principle. The formalism developed is applied to the angular expansion in polynomials of the directional flux.