1e 1985 pp.xvi+669.Uses SI units, and assumes horizontal layers of passive plasmas.It assumes a certain knowledge of electromagnetism, differential equations and tensors or at least matrices. A simpler introduction to the magneto-ionic theory is Ratcliffe The Magneto-Ionic Theory & its Applications to the Atmosphere. This book is a rewrite of the author's Radio Waves in the Ionosphere or the 2009 hbk/pbk reprint.What the book does not cover is ground wave propagation, line-of-site propagation with or without diffraction or scattering. It covers from VLF to HF.Ch.1 describes the ionosphere, a cold plasma. The Chapman layer model of the E-layer. Collision frequencies.Ch.2 Electromagnetic equations and refractive index.Ch.3 Plasma frequency. collision damping. Geomagnetic field.Ch.4 Polarisation and refractive index. Ch.5 Rays and Group Velocity.Ch 6. Inhomogenous stratified media.Ch 7. Solutions using Wentzel, Kramers and Brillouin (WKB, 1926) aka Liouville, Green (LG,1837). These are good approximations to solve a second-order differential equation. The exact solution is the Airy integral function (Ai). Budden does explain the background of the mathematics including the solution of matrix equations.Ch 8. The Airy integral function and the Stokes Phenomenon.Ch 9.Integration by Steepest Descents.Error integrals and Fresnel integrals. Contour maps. Solutions of Stokes' Equation. The method of stationary phase. Double steepest descents.Ch 10. Ray Tracing in a loss-free stratified medium.Ray path, wave packets, examples at oblique incidence, Poeverlein's construction, The Spitze, Extrordinart ray, Lateral deviation at vertical incidence, east and west, calculation of attenuation; ray pencils, caustics; cusps and catastrophes; edge focusing.Ch 11. Reflection and transmission coefficients.Reflection coefficient matrix; wave impedance and admittance; Brewster angle; shape of radio pulse.Ch 12. Ray theory results for isotropic ionosphere (Y=0).Effect of collisions on phase and equivalent height; parabolic layer; oblique paths; MUF.Ch 13. Ray theory for anisotropic plasmas (Y>1).Equivalent height; ionograms; topside sounding; Faraday rotation; whistlers; absorption.Ch 14. General Ray Tracing.Iekonal function; canonical equations; Fermat's principle; complex rays; reciprocity.The author defines Snell's law using incident angles but then uses normal angles causing some onfusion with some equations.Ch 15. Full-wave solutions for isotropic ionosphere.Linear models; vertical polarisation; electron height distributions.Ch 16. Coupled Wave Equations.Coupling and reflection points; phase integral method; Z-trace; variation of parameters.Ch 17. Coalescence of coupling points.Further matrix theory; Coalescence of 2 kinds; radio windows.Ch 18. Full-wave methods for anisotropic stratified media.LF and VLF; Runge-Kutta-Merson method;.Ch 19. Application of full-wave methods.Vertical magnetic field; resonance and barriers; solutions at higher frequencies.Every chapter has maths problems and many solutions are given. As well as a full bibliography there is an Index of definitions of the more important symbols. Some mathematical operators are missing such as * : ~ and ^.