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We find solutions of Laplace's equation with specific boundary conditions (inwhich such solutions take either the value zero or unity in each surface) usinga generic curvilinear system of coordinates. Such purely geometrical solutions(that we shall call Basic Harmonic Functions BHF's) are utilized to obtain amore general class of solutions for Laplace's equation, in which the functionstake arbitrary constant values on the boundaries. On the other hand, the BHF'sare also used to obtain the capacitance of many electrostatic configurations ofconductors. This method of finding solutions of Laplace's equation andcapacitances with multiple symmetries is particularly simple, owing to the factthat the method of separation of variables becomes much simpler under theboundary conditions that lead to the BHF's. Examples of application in complexsymmetries are given. Then, configurations of succesive embedding of conductorsare also examined. In addition, expressions for electric fields between twoconductors and charge densities on their surfaces are obtained in terms ofgeneralized curvilinear coordinates. It worths remarking that it is plausibleto extrapolate the present method to other linear homogeneous differentialequations.

Solutions of Laplace's equation with simple boundary conditions, and their applications for capacitors with multiple symmetries