  In this paper, we investigate the effects of the quantity $\sigma_{TS}$ on
the spin-structure functions of nucleons in the resonance region. The Schwinger
sum rule for the spin structure function $g_2(x,Q^2)$ at the real photon limit
is derived for the nucleon treated as a composite system, and it provides a
crucial constraint on the longitudinal transition operator which has not been
treated consistently in the literature. The longitudinal amplitude $S_{\frac
12}$ is evaluated in the quark model with the transition operator that
generates the Schwinger sum rule. The numerical results of the quantity
$\sigma_{TS}$ are presented for both spin structure functions $g_1(x,Q^2)$ and
$g_2(x,Q^2)$ in the resonance region. Our results show that this quantity
  plays an important role in the low $Q^2$ region, which can be tested in the
future experiments at CEBAF.
