The theory for the effect of air drag on satellite orbits was developed in Parts I to V of this series of papers on the assumption that the angular motion of perigee is controlled by gravitational forces and is not affected by air drag. If the orbital eccentricity is less than about 0.01, however, and the atmosphere exhibits a substantial day-to-night variation in density, the air drag itself significantly affects the angular motion of perigee. In these circumstances the mathematical theory takes a different and more complicated form, which is developed in the present paper. General equations are derived for the rates of change of parameters specifying the eccentricity and argument of perigee. Complete analytical solutions for the time variations of these parameters are obtained when e is of order 0.001, in two different forms, for (1) high drag, i.e. short lifetime, and (2) low drag. The results are illustrated by numerical examples.