Capacitance relations for a class of two-dimensional conductor configurations

A relation, discovered during the 1950s, between the two-dimensional capacitances of four almost touching shells with an arbitrary shape is revisited. A short proof of this relation is presented based on Schwarz-Christoffel conformal mapping. The relation is then applied to find closed form expressions for the capacitances in configurations of more than four almost touching conductors. The results are important for making precise capacitors, capacitor benches and as benchmark results for testing capacitance software