The investigation is based on a system of elements, obtained from the best observations made in England during the maximum of the meteor shower 1866. This system represents the most probable orbit of that part of the stream, which passed perihelion during the years 1864–67 and was later given the name of Ortho-Leonids. Seventeen points (meteor groups) have been chosen on this orbit, and differential equations of their motion have been integrated on the electronic computer BESM-2 by Cowell's method of quadratures, taking account of perturbations from eight planets (Venus–Pluto), with a variable step from 0·001 to 40 days and taking account of differences through to the 4th order. The motions of two groups (XI and XII) have been investigated in an interval of 300 years (1700–2000), and the motions of the rest of the groups for a space of 135 years (1866–2000). All the close approaches of these groups to the Earth, Jupiter, Saturn and Uranus have been determined. The results of integration are given in the tables, which clearly represent the evolution of the orbit of every group. It has been found that the basic factors determining the evolution of separate groups, and of the stream as a whole, are the close approaches to the outer planets. The perturbations by these planets, especially by Jupiter and Saturn, determine the conditions for an encounter of the meteor groups with the Earth, and cause a change in the activity of the Leonid shower at different apparitions. Apparitions of the Leonids have been investigated for the last millennium, and in more detail for the last 180 years. It has been stated that the orbit of the Ortho-Leonid stream remained stable over the interval of 1000 years; its stability has been confirmed by calculations during the last 300 years, and the limits of changes of its elements have been computed. The perturbing influence of the Earth on the motion of meteor bodies in its sphere of action has been investigated. It turns out that at exceptionally deep penetrations of meteor bodies into this sphere of action, at a distance of some 1000 km from the Earth, its perturbations can essentially transform the orbit of a meteor body, e.g. reduce its period of revolution by some years, and materially change the eccentricity of the orbit, its inclination, etc. Conditions for an encounter of the stream with the Earth in the period 1898–2000 have been clarified, and forecasts have been made for the times of maximum activity of the shower in the years 1966–68. In 1967 the maximum activity of the Leonids is predicted to occur from November 17, 18h to November 18, 1h UT.