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Using $e^+e^-$ annihilation data corresponding to an integrated luminosity of3.19\,fb$^{-1}$ collected at a center-of-mass energy of 4.178~GeV with theBESIII detector, we measure the absolute branching fractions$\mathcal{B}_{D_s^+ \rightarrow \eta e^{+} \nu_e }$ = $(2.323\pm0.063_{\rmstat}\pm0.063_{\rm syst})\%$ and $\mathcal{B}_{D_s^+ \rightarrow \eta^{\prime}e^{+} \nu_e}$ = $(0.824\pm0.073_{\rm stat}\pm0.027_{\rm syst})\%$ via a taggedanalysis technique, where one $D_s$ is fully reconstructed in a hadronic mode.Combining these measurements with previous BESIII measurements of$\mathcal{B}_{D^+\to\eta^{(\prime)} e^{+} \nu_e}$, the $\eta-\eta^\prime$mixing angle in the quark flavour basis is determined to be $\phi_{\rm P} =(40.1\pm2.1_{\rm stat}\pm0.7_{\rm syst})^\circ$. From the first measurements ofthe dynamics of $D^+_s\to \eta^{(\prime)}e^+\nu_e$ decays, the products of thehadronic form factors $f_+^{\eta^{(\prime)}}(0)$ and theCabibbo-Kobayashi-Maskawa matrix element $|V_{cs}|$ are determined withdifferent form factor parameterizations. For the two-parameter seriesexpansion, the results are $f^{\eta}_+(0)|V_{cs}| = 0.4455\pm0.0053_{\rmstat}\pm0.0044_{\rm syst}$ and $f^{\eta^{\prime}}_+(0)|V_{cs}| =0.477\pm0.049_{\rm stat}\pm0.011_{\rm syst}$.

Measurement of the Dynamics of the Decays ${ D_s^+ \rightarrow \eta^{(\prime)} e^{+} \nu_e}$