A Mean Value Theorem in Potential Theory

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Abstract
Ransford [1970] discusses the flow through a rectangular domain of a conductor with sinks or sources located on the center line between two of the sides that are streamlines, and shows that the mean potential measured transversally varies linearly from one sink or source to the next. This is a particular case of a much more general mean value theorem that is applicable to any distribution of sources and sinks of any shape in the rectangular domain and is not restricted to the small circular sources and sinks located on the center line as required in Ransford's proof by the method of images. The more general theorem may be stated as follows:For any arrangement of sources and sinks of any shape disposed between straight parallel boundaries which are themselves streamlines, the mean value of the potential function evaluated transversally across the strip at right angles to the parallel boundaries varies linearly between one source or sink and the next.

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