This paper presents a novel contribution to the analysis of skin effect phenomena in inhomogeneous tubular conductors. For homogeneous tubular geometries the skin effect diffusion equation has an analytical solution described by a combination of Bessel functions, but, when the conductivity and magnetic permeability of the tubular conductor arbitrarily depend on the radial coordinate an analytical solution cannot be found. However, this work shows that simple closed form solutions for the electromagnetic field and conductor internal impedance do exist, provided that a specific type of radial variation of medium parameters is properly chosen —these novel structures are coined here Euler-Cauchy Structures. Analytic and computation results concerning general and particular Euler-Cauchy Structures are presented, validated, and discussed. Future research on this seminal topic may lead to new engineering applications.