  Salmon's nearly geostrophic model for rotating shallow-water flow is derived
in full spherical geometry. The model, which results upon constraining the
velocity field to the height field in Hamilton's principle for rotating
shallow-water dynamics, constitutes an important prototype of Hamiltonian
balanced models. Instead of Salmon's original approach, which consists in
taking variations of particle paths at fixed Lagrangian labels and time, Holm's
approach is considered here, namely variations are taken on Lagrangian particle
labels at fixed Eulerian positions and time. Unlike the classical
quasigeostrophic model, Salmon's is found to be sensitive to the differences
between geographic and geodesic coordinates. One consequence of this result is
that the $\beta $ plane approximation, which is included in Salmon's original
derivation, is not consistent for this class of model.
