Let
x
:
M
→
S
n
+
1
(
1
)
x:M\to \mathbb {S}^{n+1}(1)
be an
n
n
-dimensional compact hypersurface with constant scalar curvature
n
(
n
−
1
)
r
,
r
≥
1
n(n-1)r,~r\geq 1
, in a unit sphere
S
n
+
1
(
1
)
,
\mathbb {S}^{n+1}(1),
n
≥
5
n\geq 5
, and let
J
s
J_s
be the Jacobi operator of
M
M
. In 2004, L. J. Alías, A. Brasil and L. A. M. Sousa studied the first eigenvalue of
J
s
J_s
of the hypersurface with constant scalar curvature
n
(
n
−
1
)
n(n-1)
in
S
n
+
1
(
1
)
,
n
≥
3
\mathbb {S}^{n+1}(1),~n\geq 3
. In 2008, Q.-M. Cheng studied the first eigenvalue of the Jacobi operator
J
s
J_s
of the hypersurface with constant scalar curvature
n
(
n
−
1
)
r
,
r
>
1
n(n-1)r, r>1
, in
S
n
+
1
(
1
)
\mathbb {S}^{n+1}(1)
. In this paper, we study the second eigenvalue of the Jacobi operator
J
s
J_s
of
M
M
and give an optimal upper bound for the second eigenvalue of
J
s
J_s
.