Following a previous study by A. Weinberg, the author investigates periodical diffusion phenomena produced in a spherical cell by a simple coupled set of chemical reactions. The general solution even for a spherical cell does not possess spherical symmetry. It is found that periodic oscillations are possible with a frequency spectrum determined by a set of “eigenvalues”. However, these oscillations are all damped even if the system of coupled reactions which is responsible for them has non-damped solutions. Therefore, although very complex and highly asymmetrical configurations of concentrations may be thus produced in the cell, none of those configurations, except some possible centrally symmetric ones, is lasting.