Studying magnetically conjugate phenomena at very high latitudes, requires a magnetic coordinate system that is smooth and well defined at the geographic poles. In addition, it should provide for accurate comparisons at different altitudes. In this report we present a variation on the corrected geomagnetic coordinate system that is well defined and smooth over the entire globe. It provides an analytic expression relating geographic coordinates, including altitude, to the magnetic coordinates. The coordinate system is produced by tracing magnetic field lines using the IGRF85 reference magnetic field model with time derivatives updating the model to 1988. An expression of the relationship in terms of spherical harmonics has been determined, which then provides the required well-defined and smooth relationship over the entire globe. Independent expansions for different altitudes show a smooth functional relationship of the coefficients of the expansion with altitude, and therefore simple interpolation schemes can be used to provide an appropriate exansion at any altitude between O km and approximately 600 km. By reversing the process, the inverse expansions relating the magnetic coordinates to geographic coordinates have also been determined. The effects of the seasonal variation in the Sun's declination along with the variation in the Sun's apparent position due to the eccentricity of the Earth's orbit result in a variation of nearly 1 hour of magnetic local time for a fixed UT over the cours e of a year. In many applications this variation may be important and should be included when presenting data in terms of magnetic latitude and MLT. ¿ American Geophysical Union 1989