Mayer has proposed a variation method of finding the two-body reduced density matrix ρ2 for the ground state of a many fermion system. In this paper it is shown explicitly that his variation method is imperfect and sometimes leads to quite unsatisfactory results. Further, his trial form of ρ2 for electron gas is translated into that for the π electron system of benzene molecule and it is shown that such a trial form is unsatisfactory in several respects.