The equation of the parabolic ionosphere model is modified in a manner which introduces very slight changes in the electron-density distribution. Using this new "quasi-parabolic" (or QP) model, it is possible to derive exact expressions describing radio-ray trajectories without introducing the approximations which are needed when dealing with the parabolic ionosphere, although magnetoionic effects are not included. The consequent accuracy of computed rays is useful as an aid in evaluating the accuracy of more versatile ray-calculation methods. Also, the exact ray expressions can be of direct !lse in calculating slight differences between rays which are nearly identical, for in this process, small inaccuracies can become magnified.