The preceding chapter was kept brief because it deals with matters of detailed technique rather than with general ideas. However, these techniques are quite essential to the use of vector analysis at the point where it merges into potential theory, and there is no limit to the illustration that might have been chosen. The following examples serve on the one hand to show that the general idea of using curvilinear systems is not in practice only realisable in the two types of polar coordinates defined in the text and, on the other, to demonstrate the power of precisely these polar coordinates. In this latter function what is presented is a selection of applications to potential theory – a selection which reflects the author's individual preference. It is far from a substitute for a systematic text on potential theory.

The reader should note that at this point there has occurred an inevitable ‘clash of symbols‘. In vector analysis ϕ normally stands for a scalar potential. In polar coordinates it denotes an azimuth angle. In this section only the letter ψ rather than ϕ has been used to denote all scalar potential functions that occur.