  Based on an improved U(1) slave-boson approach to the t-J Hamiltonian, we
invest igate a relationship between the superfluid weight $n_s/m^*$(the
superconducting charge carrier density/the effective mass of the charge
carrier) and the superc onducting temperature $T_c$. From the present study we
find a linear increase of $n_s/m^*$ with $T_c$ with th e doping concentration
in the underdoped region, a saturation around the optimal doping and a decrease
in both $n_s/m^*$ and $T_c$ in the overdoped region. Such a trend of the
`boomerang' shaped locus in $n_s/m^*$ vs. $T_c$ with increas ing doping
concentration from the underdoped to the heavily overdoped region is predicted
to be in complete agreement with muon-spin-relaxation measurements. The
boomerang behavior is found to occur in correlation with reduction in the sp in
singlet pairing(spinon pairing) order in the heavily overdoped region.
