For helium and helium-like ions, we have examined the differences between the values of the ionization energies as calculated from the Bohr theory and those measured in experiments. We find that these differences vary linearly with the atomic number of the system. Using this result, we show how the Bohr model for single-electron systems may be modified in a simple way so that it may be applied to two-electron systems as well. (Contains 3 figures and 1 table.)