Maxwell and the boundary element method: a historical puzzle

The boundary element method (BEM) is an important technique used for calculating the electric field around conductors and in insulator-conductor systems. It is also known as the surface charge simulation method (SCSM), and occasionally as the boundary integral element method (BIE, BIM, BIEM). Recent applications have stressed how easily it handles open boundaries. Before the term finite element came into favour, it was known as the method of moments or the method of sub-areas. Maxwell appears to have been the first to use the method. Despite pioneering the method, Maxwell was not content to do things in the simplest possible way. Unhappily, there appear to be a surprising number of inconsistencies in Maxwell's work, which we will investigate in this paper.