= Solution
{c}
The <pressure> adjustment from part (i) can be written
$$
\boxed{\mathbf T=\mathbf T_0(t)+q(\mathbf x,t)\mathbf1,\qquad
q=-\rho\xi+C(t).}
$$
Indeed, since $\operatorname{div}(q\mathbf1)=\nabla q$, the difference of the two <Cauchy momentum equations> is
$$
\nabla q=\rho\ddot{\mathbf x}_0=-\rho\nabla\xi.
$$
Thus $q+\rho\xi$ is constant in space on each connected component. The function $C(t)$ is fixed by any prescribed <pressure> or surface <traction> datum. In terms of the usual compressive <pressure> multiplier, $p=p_0-q=p_0+\rho\xi-C(t)$. The plus sign in $q\mathbf1$ therefore does not identify $q$ with the compressive multiplier $p$ itself.
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