  We derive a new relation between the observed Lamb shift energies of hydrogen
and muonium atoms. The relation is based on the non-relativistic description of
the Lamb shift, and the proper treatment of the reduced mass of electron and
target particles (proton and muon) leads to the new formula which is expressed
as $\displaystyle{{\Delta E^{(H)}_{2s_{1/2}}\over \Delta E^{(\mu)}_{2s_{1/2}}}
=({1+{m_e\over m_\mu}\over 1+{m_e\over M_p}})^3}$. This relation achieves an
excellent agreement with experiment and presents an important QED test free
from the cutoff momentum $\Lambda$.
