  In the small $Q^2$ region, the sum rule for the polarized structure function
$g_2$ corresponding to the moment at $n=0$ is derived. This sum rule shows that
there is a tight connection among the resonances,the elastic and the continuum
in the $g_2$. Further, the Born term contribution in this sum rule is
proportional to $Q^2$ and very small compared with that in the corresponding
sum rule for the polarized structure function $g_1$. However, the Born term
contribution divided by $Q^2/2$ which also appears in the Schwinger sum rule
for the $g_2$ corresponding to the moment at $n=1$ has a very similar behavior
with that in the sum rule for the $g_1$ corresponding to the moment at $n=0$.
