  This paper presents results from partial-wave analyses of the photoproduction
reactions $\gamma p \rightarrow \eta p$ and $\gamma n \rightarrow \eta n$.
World data for the observables \DSG, $\Sigma$, $T$, $P$, $F$, and $E$ were
analyzed as part of this work. The dominant amplitude in the fitting range from
threshold to a c.m.\ energy of 1900 MeV was found to be $S_{11}$ in both
reactions, consistent with results of other groups. At c.m.\ energies above
1600 MeV, our solution deviates from published results, with this work finding
higher-order partial waves becoming significant. Data off the proton suggest
that the higher-order terms contributing to the reaction include $P_{11}$,
$P_{13}$, and $F_{15}$. The final results also hint that $F_{17}$ is needed to
fit double-polarization observables above 1900 MeV. Data off the neutron show a
contribution from $P_{13}$, as well as strong contributions from $D_{13}$ and
$D_{15}$.
