For particles crossing the orbit (2) the mean velocity would be practically parabolic, 1.4, even if the distribution were as 1/_r_′, _r_′ being the distance from the Sun. The effect would depend upon the angle of approach and in the mean give a greater velocity for the particle than for the satellite within the orbit, a less one without; retarding the satellite in both cases. Thus the total effect of all the particles encountering the large satellites is to retard them and to tend to make them hug their primary. For retrograde satellites the velocities of impact with inside and outside particles moving direct are respectively: =========+===========+========== | INSIDE | OUTSIDE ---------+-----------+--------- Jupiter | 2.0 + _v_ | _v_ + 3.4 Saturn | 1.5 + _v_ | _v_ + 2.5 Uranus | 1.0 + _v_ | _v_ + 1.7 Neptune | 0.8 + _v_ | _v_ + 1.4 =========+===========+========= In both cases the impact tends to check the satellite. Comparing with these the velocities of impact for direct satellites in a direct plenum:— =========+===========+=========== | INSIDE | OUTSIDE ---------+-----------+----------- Jupiter | 2.0 - _v_ | 3.4 - _v_ Saturn | 1.5 - _v_ | 2.5 - _v_ Uranus | 1.0 - _v_ | 1.7 - _v_ Neptune | 0.8 - _v_ | 1.4 - _v_ =========+===========+=========== the signs being taken positive when the motion is direct, we see that retrograde satellites would be more arrested than direct ones with the same orbital speed round the primary. In a plenum of direct moving particles, then, the force tending to stop the satellite and bring it down upon the planet is greater for retrograde satellites than for direct ones. If, therefore, the positions of the satellites have been controlled by the impact of interplanetary particles, the retrograde satellites should be found nearer their planets than the direct ones. 6 ON THE INDUCED CIRCULARITY OF ORBITS THROUGH COLLISION Since the moment of momentum is the velocity into the perpendicular upon its direction, in the time _dt_ it is:— _vp dt_ = _h dt_ = _r_²_d_Θ. The whole moment of momentum from perihelion to perihelion is therefore:— ⌠360° │ _r_²_d_Θ = _a_²·(1-_e_²)²/1-_e_² ⌡₀ ┌360° │ (-_e_ sin Θ)/(1+_e_ cos Θ) └₀ _____________ ┐ + 2/(1-_e_²)^{½} tan⁻¹ (√1-_e_)/(1+_e_·tan (Θ/2))│ ┘ = 2π_a_² · (1 - _e_²)^{½}, which is twice the area of the ellipse. The energy in the ellipse during an interval _dt_ is (½)_mv_²_dt_ = (½)_m_µ(2/_r_ - 1/_a_)_dt_, from the well-known equation for the velocity in a focal conic. The integral of this for the whole ellipse is ⌠ᵀ ⌠360° │ (½)_mv_² _dt_ = │ (½)(_m_µ/_h_)(2_r_ - _r_²/_a_)_d_Θ ⌡₀ ⌡₀ = _m_µ^{½}π_a_^{½}. Since