We prove two mathematically rigorous theorems that assert, under certain carefully stated hypotheses, the validity of the Goertzel and Otsuka conclusions that, in a thermal nuclear reactor that has a minimum critical mass, the fuel must be distributed so that the product of the thermal neutron flux and the adjoint thermal neutron flux is a constant in the core and does not exceed that constant in the reflector. These theorems differ from that in the preceding paper in the sense that some of the hypotheses of the earlier theorem have been strengthened and some weakened. The hypotheses can be weakened still further if we restrict attention to a fixed core and are not interested in results concerning the reflector. We also study the second variation of the critical mass functional. Finally, we show that, under some explicitly stated conditions, the multigroup diffusion theory for a thermal reactor can be treated as a special case of our general theory.