= Solution
Resolve <gravity> into components $g\sin\alpha$ down the bed and $-g\cos\alpha$ normal to it. For slow <incompressible flow>, <inertia> is negligible and <Cauchy momentum equation> reduces to the divergence of the <stress tensor> plus the gravitational <force density> being zero. Write the dominant shear component as $\tau=\sigma_{xz}$ and the leading normal stresses as $-p$.
The <shallow-ice approximation> assumes a small thickness-to-length ratio. Vertical variation of the shear is retained in the along-bed equation, while longitudinal deviatoric-stress gradients and the shear contribution to the normal equation are higher order. The leading balances are consequently
$$
\boxed{-p_x+\tau_z+\rho g\sin\alpha=0,\qquad -p_z-\rho g\cos\alpha=0.}
$$
The second is a <hydrostatic pressure> balance; the first balances gravitational driving and the surface-induced <pressure gradient> against vertical shear. This approximation requires shallow geometry and sufficiently slow flow, not a Newtonian <constitutive equation> for <ice>.
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