Table of contents : Title Introduction and Summary General Considerations On the Wave Functions of the Parabolic Layer On the Expansions of the Parabolic Wave Functions The Transmission of Radio Waves round a Spherical Earth surrounded by a Radially Inhomogeneous Concentric Reflecting Shell The Transformation of the Series The Poles of the Watson Case The Numerical Evaluation of the Poles The Residue Series The Treatment of the Separate Wave Groups The Field Strength from the Dipole Element The Reflection Coefficient of the Parabolic Layer On the Application of the Phase Integral The Boundary Reflection of the Parabolic Layer A Short Note on the Reflection Coefficient of a Layer with Quadratic Increase in Electron Density A Short Discussion of the Case ωcm