It is the purpose of this paper to develop a theory analogous, and, roughly speaking, dual to Kasner's recent theory of polygenic functions of z. There seems to be no general principle of transference whereby the properties of the functions here considered could be directly inferred from the corresponding facts in the original theory. A summary of some of the results obtained below will serve to illustrate points of similarity and difference in the two theories.
The first derivative of a polygenic function of z is represented by a congruence of circles. For functions of the dual variable we have a congruence of directed circles, or cycles. In the matter of the method of generation of these curves, the theories diverge. Kasner has found it appropriate, with this question in view, so speak of a congruence of clocks. It will be shown below that our derivative cycles are described in an entirely different way. There is also some variation in the nature of the specialization of the congruences. For the second derivative we have as corresponding pictures, a line and a cycle, a curve of eighth order and a directed curve of sixth order.