  Based on a sample of $1.31 \times 10^9$ $J/\psi$ events collected with the
BESIII detector at the BEPCII collider, Dalitz plot analyses of selected 79,625
$\eta\rightarrow\pi^{+}\pi^{-}\pi^0$ events, 33,908
$\eta\rightarrow\pi^0\pi^0\pi^0$ events and 1,888
$\eta^{\prime}\rightarrow\pi^0\pi^0\pi^0$ events are performed. The measured
matrix elements of $\eta\rightarrow\pi^+\pi^-\pi^0$ are in reasonable agreement
with previous measurements. The Dalitz plot slope parameters of
$\eta\rightarrow\pi^0\pi^0\pi^0$ and $\eta^{\prime}\rightarrow\pi^0\pi^0\pi^0$
are determined to be $-0.055 \pm 0.014 \pm 0.004$ and $-0.640 \pm 0.046 \pm
0.047$, respectively, where the first uncertainties are statistical and the
second systematic. Both values are consistent with previous measurements, while
the precision of the latter one is improved by a factor of three. Final state
interactions are found to have an important role in those decays.
