The spatial distribution of the scalar flux is investigated for vanishinly small critical systems in the three standard one dimensional geometries. In slab geometry, this distribution is found to be flat. In spherical and cylindrical systems, one finds the unexpected result that the scalar flux distributions do not approach a constant as the radius of the system goes to zero. Numerical results are given for the scalar flux distributions in this vanishingly small limit. Numerical results are also given for the criticality conditions for these three geometries for small, but non-zero size, systems. These exact results are used to re-investigate the criticality problem for highly anisotrbpic scattering. The trends previously reported from an approximate variational approach are confirmed. An argument is presented why the standard reasoning which leads to the conclusion that vanishing small systems have spatially flat flux distribution is not valid for critical systems.