Relative Decay Rates of $\eta \to {\pi }^{+}{\pi }^{-}{\pi }^0$, $\eta \to {\pi }^{+}{\pi }^{-}\gamma$, $\eta \to {\pi }^{+}{\pi }^{-}{\pi }^0\gamma$ and $\eta \to {\pi }^{+}{\pi }^{-}$

We have measured the branching ratio $\frac{(\eta \to {\pi }^{+}{\pi }^{-}\gamma )}{(\eta \to {\pi }^{+}{\pi }^{-}{\pi }^0)}$ to be 0.209±0.004. The rare decays $\eta \to {\pi }^{+}{\pi }^{-}{\pi }^0\gamma$ and $\eta \to {\pi }^{+}{\pi }^{-}$ were searched for and not found. The upper limit on the branching ratio $\frac{(\eta \to {\pi }^{+}{\pi }^{-}{\pi }^0\gamma )}{(\eta \to {\pi }^{+}{\pi }^{-}{\pi }^0)}$ is 0.0024 at 90% confidence level, and the upper limit on the width of the decay $\eta \to {\pi }^{+}{\pi }^{-}$ is 4 eV. We then have a limit on the rate of a $P$- and $T$-violating strong interaction: $\frac{\mathrm{Rate} (P-\mathrm{and} T-\mathrm{violating}\mathrm{strong}\mathrm{interaction})}{\mathrm{Rate} \left(\mathrm{conventional}\mathrm{strong}\mathrm{interaction}\right)}<4\mathrm{}\times {10}^{-8\mathrm{}}.$