  The decay $D^{+}_{s}\rightarrow \pi^{+}\pi^{+}\pi^{-}\eta$ is observed for
the first time, using $e^+e^-$ collision data corresponding to an integrated
luminosity of 6.32 fb$^{-1}$, collected by the BESIII detector at
center-of-mass energies between 4.178 and 4.226 GeV. The absolute branching
fraction for this decay is measured to be $\mathcal{B}(D^+_s \to \pi^+ \pi^+
\pi^- \eta) = (3.12\pm0.13_{\rm stat.}\pm0.09_{\rm syst.})$%. The first
amplitude analysis of this decay reveals the sub-structures in
$D^{+}_{s}\rightarrow \pi^{+}\pi^{+}\pi^{-}\eta$ and determines the relative
fractions and the phases among these sub-structures. The dominant intermediate
process is $D^{+}_{s}\rightarrow a_1(1260)^+ \eta, a_1(1260)^+ \rightarrow
\rho(770)^0\pi^+$ with a branching fraction of $(1.73 \pm 0.14_{\rm stat.} \pm
0.08_{\rm syst.})$%. We also observe the W-annihilation process
$D^{+}_{s}\rightarrow a_0(980)^+\rho(770)^0$, $a_0(980)^+ \to \pi^+ \eta$ with
a branching fraction of $(0.21\pm0.08_{\rm stat.}\pm0.05_{\rm syst.})$%, which
is larger than the branching fractions of other measured pure W-annihilation
decays by one order of magnitude.
