Beam asymmetry <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="normal">Σ</mml:mi></mml:math> for the photoproduction of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>η</mml:mi></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>η</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:math> mesons at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>E</mml:mi><mml:mi>γ</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>8.8</mml:mn><mml:mspace width="0.28em"/><mml:mi>GeV</mml:mi></mml:mrow></mml:math>

We report on the measurement of the beam asymmetry $\mathrm{\ensuremath{\Sigma}}$ for the reactions $\stackrel{P\vec}{\ensuremath{\gamma}}p\ensuremath{\rightarrow}p\ensuremath{\eta}$ and $\stackrel{P\vec}{\ensuremath{\gamma}}p\ensuremath{\rightarrow}p{\ensuremath{\eta}}^{\ensuremath{'}}$ from the GlueX experiment using an 8.2--8.8-GeV linearly polarized tagged photon beam incident on a liquid hydrogen target in Hall D at Jefferson Laboratory. These measurements are made as a function of momentum transfer $\ensuremath{-}t$ with significantly higher statistical precision than our earlier $\ensuremath{\eta}$ measurements and are the first measurements of ${\ensuremath{\eta}}^{\ensuremath{'}}$ in this energy range. We compare the results to theoretical predictions based on $t$-channel quasiparticle exchange. We also compare the ratio of ${\mathrm{\ensuremath{\Sigma}}}_{\ensuremath{\eta}}$ to ${\mathrm{\ensuremath{\Sigma}}}_{{\ensuremath{\eta}}^{\ensuremath{'}}}$ to these models as this ratio is predicted to be sensitive to the amount of $s\overline{s}$ exchange in the production. We find that photoproduction of both $\ensuremath{\eta}$ and ${\ensuremath{\eta}}^{\ensuremath{'}}$ is dominated by natural parity exchange with little dependence on $\ensuremath{-}t$.