An electrostatic problem in bi-cyclide coordinates

This paper deals with an electrostatic problem for the field between two charged conductors $\pm {u_0}$ maintained at potential $\pm V$ in bi-cyclide coordinates ($u, v, \psi$). In this coordinate system, Heine functions are used, of which something is known. Heine functions are the solutions of Heine differential equations. Though the problem is to be solved in the same manner as the problem in the case of bispherical coordinates, it has not been clarified because of the complexity of Heine functions. A Heine differential equation is solved to satisfy the boundary condition that the functions and their derivatives are bounded at the ends of interval, and eigenvalues and eigenfunctions are evaluated. A formula giving the capacity between two electrodes is presented and numerically calculated.