Measurements of the absolute branching fractions for $D_{s}^{+}rightarroweta e^{+} u_{e}$ and $D_{s}^{+}rightarroweta^{prime} e^{+} u_{e}$

By analyzing 482 pb$^{-1}$ of $e^+e^-$ collision data collected at $sqrt s=4.009$ GeV with the BESIII detector at the BEPCII collider, we measure the absolute branching fractions for the semileptonic decays $D_{s}^{+}toeta e^{+} u_{e}$ and $D_{s}^{+}to etae^{+} u_{e}$ to be ${B}(D_{s}^{+}rightarroweta e^{+} u_{e})=(2.30pm0.31pm0.08)$% and ${B}(D_{s}^{+}rightarrowetae^{+} u_{e}) = (0.93pm0.30pm0.05)$%, respectively, and their ratio $frac{{B}(D_{s}^{+}rightarrowetae^{+} u_{e})} {{B}(D_{s}^{+}rightarroweta e^{+} u_{e})}=0.40pm0.14pm0.02$, where the first uncertainties are statistical and the second ones are systematic. The results are in good agreement with previous measurements within uncertainties; they can be used to determine the $eta-eta$ mixing angle and improve upon the $D_s^+$ semileptonic branching ratio precision.