Bicylindrical coordinates

The numbers $ \tau , \sigma $ and $ z $ related to the rectangular Cartesian coordinates $ x, y $ and $ z $ by the formulas

$$ x = \frac{a \sinh \tau }{\cosh \tau - \cos \sigma } ,\ \ y = \frac{a \sin \sigma }{\cosh \tau - \cos \sigma } ,\ \ z = z, $$

where $ 0 \leq \sigma < \pi , -\infty < \tau < \infty $. The coordinate surfaces are: the family of pairs of circular cylinders with parallel axes ( $ \tau = \textrm{ const } $), the family of circular cylinders orthogonal to the former ( $ \sigma = \textrm{ const } $), and the planes ( $ z = \textrm{ const } $). The system of bicylindrical coordinates is obtained as the result of translation of the system of bipolar coordinates in the $ xy $- plane parallel to the $ z $- axis.

The Lamé coefficients (cf. Lamé coefficients) are:

$$ L _ \sigma = L _ \tau = \ \frac{a ^ {2} }{( \cosh \tau - \cos \sigma ) ^ {2} } ,\ \ L _ {z} = 1. $$

The Laplace operator is:

$$ \Delta f = \frac{1}{a ^ {2} } ( \cosh \tau - \cos \sigma ) ^ {2} \left ( \frac{\partial ^ {2} f }{\partial \sigma ^ {2} } + \frac{\partial ^ {2} f }{\partial \tau ^ {2} } \ \right ) + \frac{\partial ^ {2} f }{\partial z ^ {2} } . $$