In view of the various cosmogonies which have been advanced for the genesis of the solar system it is interesting to note what these speeds imply as to the effect upon the satellites of the impact of particles circulating in the interplanetary spaces at the time the system evolved. To simplify the question we shall suppose—which is sufficiently near the truth—that the planets move in circles, the interplanetary particles in orbits of any eccentricity. Taking the Sun’s mass as unity, the distance _R_ of any given planet from the Sun also as unity, let the planet’s mass be represented by _M_ and the radius of its satellite’s orbit, supposed circular, as _r_. We have for the space velocity of the satellite on the sunward side of the planet, calling that of the planet in its orbit _V_ and that of the satellite in its orbit round the planet _v_, _______ _______ _V_ - _v_ = √(1/_R_) - √_M_/_r_. For a particle, the semi-major axis of whose orbit is _a₁_ and which shall encounter the satellite, the velocity is _v₁_ = (2/(_R_-_r_) - 1/_a₁_)^{½}. That no effect shall be produced by the impact of these two bodies, their velocities must be equal, or _____ _______ ____________________ √1/_R_ - √_M_/_r_ = √2/(_R_-_r_) - 1/_a₁_ As _R_-_r_ = _a₁_(1 + _e_) for the point of impact if the particle be wholly within the orbit of the planet and _e_ the eccentricity of its orbit, we find _________________ _e_ = 2 √_MR_/_r_ - _RM_/_r_ approx. for the case of no action, the other terms being insensible for the satellites in the table, since in all _r_ < _R_/400. Supposing, now, the particles within the orbit of the planet to be equally distributed according to their major axes, then as the velocity of any one of them, taking _R_-_r_ = _R_ approx. as unity, is _v₁_ = (2/1 - 1/_a₁_)^{½}, the mean velocity of all of those which may encounter the satellite is, at the point of collision, ⌠¹ │ ((2_a₁_ - 1)^{½} / _a₁_^{½})_da₁_ ⌡_{½} ———————————————————————————————————— ⌠¹ │ _da₁_ ⌡_{½} ┌¹ __ _____ ┐ = 2│ (2_a₁_² - _a₁_)^{½} - 1/√(2) log{(2_a₁_ - 1)^{½} + √2_a₁_ }│ └_{½} ┘ = 0.754; that is, just over three-quarters of the planet’s speed in its orbit. If we suppose the particles to be equally distributed in space, we shall have more with a given major axis in proportion to that axis, and our integral will become ⌠¹ │ (2_a₁_ - 1)^{½}_a₁_^{½} _da₁_ ⌡_{½} ———————————————————————————————— ⌠¹ │ _a₁ da₁_ ⌡_{½}