  The spin-structure functions $g_1$ and $g_2$, and the spin-dependent partial
cross-section $\sigma_\mathrm{TT}$ have been extracted from the polarized
cross-sections differences, $\Delta
\sigma_{\parallel}\hspace{-0.06cm}\left(\nu,Q^{2}\right)$ and $\Delta
\sigma_{\perp}\hspace{-0.06cm}\left(\nu,Q^{2}\right)$ measured for the
$\vec{^\textrm{3}\textrm{He}}(\vec{\textrm{e}},\textrm{e}')\textrm{X}$
reaction, in the E97-110 experiment at Jefferson Lab. Polarized electrons with
energies from 1.147 to 4.404 GeV were scattered at angles of 6$^{\circ}$ and
9$^{\circ}$ from a longitudinally or transversely polarized $^{3}$He target.
The data cover the kinematic regions of the quasi-elastic, resonance production
and beyond. From the extracted spin-structure functions, the first moments
$\overline{\Gamma_1}\hspace{-0.06cm}\left(Q^{2}\right)$,
$\Gamma_2\hspace{-0.06cm}\left(Q^{2}\right)$ and
$I_{\mathrm{TT}}\hspace{-0.06cm}\left(Q^{2}\right)$ are evaluated with high
precision for the neutron in the $Q^2$ range from 0.035 to 0.24~GeV$^{2}$. The
comparison of the data and the chiral effective field theory predictions
reveals the importance of proper treatment of the $\Delta$ degree of freedom
for spin observables.
