Nonlinear resistive cascades that extend beyond infinity

In previous work, the author (1991) established a theory for transfinite cascades of three-terminal networks wherein the port terminals are indexed consecutively first by the natural numbers k =0, 1, 2, . . . and then by the transfinite ordinals k=ω, ω+1, ω+2, . . . up to the ordinal ω^p, where p is any natural number. Under the conditions imposed on the three-terminal networks, it will become necessary to specify a load at both the input and output of that structure in order to determine the currents and voltages throughout the cascade. For definiteness, the author shows the loads in the Thevenin form on the input side and in the Norton form on the output side, but either form can occur on either end. The resistances of those loads are also nonlinear in general. The purpose of the present work is to extend the theory to allow p to be any countable ordinal. The author first summarizes the prior result and then discusses the extension