Observation of the decays ${\chi }_{cJ}\to \varphi \varphi \eta $

Using a data sample of $\left(448.1±2.9\right)×{10}^{6}\text{}\text{}\psi \left(3686\right)$ decays collected by the BESIII detector at the Beijing Electron Positron Collider (BEPCII), we observe the decays ${\chi }_{cJ}\to \varphi \varphi \eta \left(J=0,1,2\right)$, where the ${\chi }_{cJ}$ are produced via the radiative processes $\psi \left(3686\right)\to \gamma {\chi }_{cJ}$. The branching fractions are measured to be $B\left({\chi }_{c0}\to \varphi \varphi \eta \right)=\left(8.41±0.74±0.62\right)×{10}^{-4}$, $B\left({\chi }_{c1}\to \varphi \varphi \eta \right)=\left(2.96±0.43±0.22\right)×{10}^{-4}$, and $B\left({\chi }_{c2}\to \varphi \varphi \eta \right)=\left(5.33±0.52±0.39\right)×{10}^{-4}$, where the first uncertainties are statistical, and the second are systematic. We also search for intermediate states in the $\varphi \varphi $ or $\eta \varphi $ subsystems, but no significant structure is seen due to the limited statistics.