  Let $x:M\to\mathbb{S}^{n+1}(1)$ be an n-dimensional compact hypersurface with
constant scalar curvature $n(n-1)r,~r\geq 1$, in a unit sphere
$\mathbb{S}^{n+1}(1),~n\geq 5$. We know that such hypersurfaces can be
characterized as critical points for a variational problem of the integral
$\int_MH dv$ of the mean curvature $H$. In this paper, we derive an optimal
upper bound for the second eigenvalue of the Jacobi operator $J_s$ of $M$.
Moreover, when $r>1$, the bound is attained if and only if $M$ is totally
umbilical and non-totally geodesic, when $r=1$, the bound is attained if $M$ is
the Riemannian product
$\mathbb{S}^{m}(c)\times\mathbb{S}^{n-m}(\sqrt{1-c^2}),~1\leq m\leq
n-2,~c=\sqrt{\frac{(n-1)m+\sqrt{(n-1)m(n-m)}}{n(n-1)}}$.
