$$\rho=\tan\phi \sec\phi \cos\theta.$$
Work Step by Step
We apply the equations of this section (page 699). We have $$x=z^2 \Longrightarrow \rho\sin\phi\cos\theta=\rho^2\cos^2\phi .$$ That is $$\rho=\tan\phi \sec\phi \cos\theta.$$
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