  Using a data sample of $58\times 10^6$ $J/\psi$ decays collected with the BES
II detector at the BEPC, searches for invisible decays of $\eta$ and
$\eta^\prime$ in $J/\psi$ to $\phi\eta$ and $\phi\eta^\prime$ are performed.
The $\phi$ signals, which are reconstructed in $K^+K^-$ final states, are used
to tag the $\eta$ and $\eta^\prime$ decays. No signals are found for the
invisible decays of either $\eta$ or $\eta^\prime$, and upper limits at the 90%
confidence level are determined to be $1.65 \times 10^{-3}$ for the ratio
$\frac{B(\eta\to \text{invisible})}{B(\eta\to\gamma\gamma)}$ and $6.69\times
10^{-2}$ for $\frac{B(\eta^\prime\to
\text{invisible})}{B(\eta^\prime\to\gamma\gamma)}$. These are the first
searches for $\eta$ and $\eta^\prime$ decays into invisible final states.
