  A lower bound on $\Delta m^{2}$ for atmospheric neutrino oscillations can be
obtained from the global value of $R$ at a given energy. This bound may be more
reliable than one based on the observed isotropy. Some global values of $R$
imply an anisotropy in the neutrino flux that may, or may not be manifest in
the observed neutrino interactions. A limit of $\Delta m^{2}>0.0077$ eV$^{2}$
is easily obtained. Much higher mass scales are implied by the reported energy
independence and magnitude of $R$
