ON THE NUMERICAL CALCULATION OF STATIC MULTIPOLE FIELDS.

In order to solve Poisson's equation in three dimensions the potential and the charge density can be expanded into azimuthal Fourier series the result being a set of decoupled two-dimensional differential equations. For the numerical solution of these a discretization is developed that is based upon eight-point mesh formulae in square grids, the discretization error being of sixth order in the mesh width. The resulting system of linear equations can be solved by successive overrelaxation for which an approximative theory is given. For mesh points in the vicinity of a boundary an asymmetric four-point formula is derived.