  In this paper, we study the limiting properties of the $K$ energy for smooth
hypersurfaces in the projective spaces. Our result generalizes the result of
Ding-Tian (W. Ding and G. Tian. K\"ahler-Einstein metrics and the generalized
Futaki invariant. {\em Invent Math}, 110:315-335, 1992.) in the case of
hypersurfaces. In particular, we allow the center fiber of a special
degeneration (a degeneration by a one-parameter group) to have multiplicity
great than 1.
