  We report on the measurement of the beam asymmetry $\Sigma$ for the reactions
$\vec{\gamma}p\rightarrow p\eta$ and $\vec{\gamma}p \rightarrow p\eta^{\prime}$
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 Lab.
These measurements are made as a function of momentum transfer $-t$, with
significantly higher statistical precision than our earlier $\eta$
measurements, and are the first measurements of $\eta^{\prime}$ in this energy
range. We compare the results to theoretical predictions based on $t$--channel
quasi-particle exchange. We also compare the ratio of $\Sigma_{\eta}$ to
$\Sigma_{\eta^{\prime}}$ to these models, as this ratio is predicted to be
sensitive to the amount of $s\bar{s}$ exchange in the production. We find that
photoproduction of both $\eta$ and $\eta^{\prime}$ is dominated by natural
parity exchange with little dependence on $-t$.
