= Prolate spheroidal coordinates
{title2=$(\lambda,\mu,\phi)$}
For $0<\alpha<\beta$, define $-\beta\leq\mu\leq-\alpha\leq\lambda$ by $R^2/(\tau+\alpha)+z^2/(\tau+\beta)=1$. Then $R^2=(\lambda+\alpha)(\mu+\alpha)/(\alpha-\beta)$ and $z^2=(\lambda+\beta)(\mu+\beta)/(\beta-\alpha)$. Constant $\lambda$ surfaces are confocal <prolate spheroids> and constant interior $\mu$ surfaces are two-sheeted hyperboloids, with foci $z=\pm\sqrt{\beta-\alpha}$. The coordinates are orthogonal away from their degeneracies. The squared scale factors are $h_\lambda^2=(\lambda-\mu)/[4(\lambda+\alpha)(\lambda+\beta)]$, $h_\mu^2=(\mu-\lambda)/[4(\mu+\alpha)(\mu+\beta)]$ and $h_\phi^2=R^2$. The two signs of $z$ are coordinate branches, joined at the equatorial fold $\mu=-\beta$.
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