In an earlier paper (5) we studied the inversion theory of the Gauss transformation defined by

(1.1). Operational methods indicated that and we showed that in certain circumstances this equation was true if exp (— D2) f(x) was interpreted as the sum of the series

A considerable literature has grown up around the analysis of the structure of a function in terms of its derivative, and the structure of functions F(x) which are integrals of various kinds. Some of this relates to derivatives and integrals of F(x) with respect to functions of bounded variation ω(x) (1-6) or, in the case of a paper by Ward (4), with respect to a function of generalized bounded variation in the restricted sense.

In recent years many of the properties of regular functions have been shown to extend to quasiconformal mappings. (The latter term is here understood in the sense defined in (5).) This is particularly true of those results which can be proved by use of the method of the extremal metric. It is rather strange, then, that a result which constitutes one of the first notable applications of this method has not been so extended (at least to the author's knowledge).

In acoustic and electromagnetic theory much use is made of what is called the Principle of Babinet (1). The principle states that the problems of diffraction by an aperture, S, in a plane screen and by a plane obstacle occupying the position S, are equivalent. It is the intent of this paper to indicate how this notion of equivalent boundary problems, for partial differential equations, can be extended to other situations.