Part III of this series of papers developed the theory of high-eccentricity orbits (e > 0.2) in an atmosphere having an exponential variation of air density with height, that is, with the density scale height H taken as constant. Part IV derived the appropriate theory for low-eccentricity orbits (e < 0.2) in a more realistic atmosphere where H varies linearly with height y (and $\mu$ = dH/dy < 0.2). The present paper treats the orbits of part III when they meet the air drag specified by the atmospheric model of part IV. Equations are derived showing how the perigee height varies with eccentricity, and the eccentricity varies with time, over the major part of the satellite's life. It is shown that the theory of part III remains valid, to order $\mu^2$, if H is evaluated at a specific height above perigee.