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Let $x:M\to\mathbb{S}^{n+1}(1)$ be an n-dimensional compact hypersurface withconstant 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 becharacterized as critical points for a variational problem of the integral$\int_MH dv$ of the mean curvature $H$. In this paper, we derive an optimalupper 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 totallyumbilical and non-totally geodesic, when $r=1$, the bound is attained if $M$ isthe Riemannian product$\mathbb{S}^{m}(c)\times\mathbb{S}^{n-m}(\sqrt{1-c^2}),~1\leq m\leqn-2,~c=\sqrt{\frac{(n-1)m+\sqrt{(n-1)m(n-m)}}{n(n-1)}}$.

Second eigenvalue of a Jacobi operator of hypersurfaces with constant scalar curvature