  In this paper, we give an explicit second variation formula for a biharmonic
hypersurface in a Riamannian manifold similar to that of a minimal
hypersurface. We then use the second variation formula to compute the stability
index of the known biharmonic hypersurfaces in a Euclidean sphere, and to prove
the non-existence of unstable proper biharmonic hypersurface in a Euclidean
space or a hyperbolic space, which adds another special case to support Chen's
conjecture on biharmonic submanifolds.
