= Solution
Let $\mathbf v$ denote the velocity relative to the uniformly rotating frame, and write $r_\perp^2=x^2+y^2$. For rotation $\boldsymbol\Omega=\Omega\mathbf e_z$, the <Euler equations for an inviscid fluid> in that frame are
$$
\partial_t\mathbf v+(\mathbf v\cdot\nabla)\mathbf v+2\boldsymbol\Omega\times\mathbf v=-\nabla\left(\frac p\rho+\Phi-\tfrac12\Omega^2r_\perp^2\right).
$$
The negative quadratic term is the <centrifugal potential>, whose negative gradient is the outward <centrifugal acceleration> per unit mass. In <steady flow>, dotting the equation with $\mathbf v$ eliminates the <Coriolis force>, since $\mathbf v\cdot(\boldsymbol\Omega\times\mathbf v)=0$. The remaining acceleration term is $\mathbf v\cdot\nabla(|\mathbf v|^2/2)$. Thus
$$
\mathbf v\cdot\nabla\left(\tfrac12|\mathbf v|^2+\frac p\rho+\Phi-\tfrac12\Omega^2r_\perp^2\right)=0.
$$
By part (b), \b[the <steady rotating-frame Bernoulli integral> is]
$$
\boxed{\tfrac12|\mathbf v|^2+\frac p\rho+\Phi-\tfrac12\Omega^2(x^2+y^2)=\text{constant on each relative streamline}.}
$$
Steadiness here is measured in the rotating frame, including the body-force potential used in that frame.
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