one for each of the elements.
The general equations expressing the motion of a planet considered as a material particle round a centre of attraction lead to theorems the more interesting of which will now be enunciated.
(1) The motion of such a planet may take place not only in an ellipse but in any curve of the second order; an ellipse, hyperbola, or parabola, the latter being the bounding curve between the other two. A body moving in a parabola or hyperbola would recede indefinitely from its centre of motion and never return to it. The ellipse is therefore the only closed orbit.
(2) The motion takes place in accord with Kepler’s laws, enunciated elsewhere.
(3) Whewell’s theorem: if a point R be taken at a distance from the sun equal to the major axis of the orbit of a planet and, therefore, at double the mean distance of the planet, the speed of the latter at any point is equal to the speed which a body would acquire by falling from the point R to the actual position of the planet. The speed of the latter may, therefore, be expressed as a function of its radius vector at the moment and of the major axis of its orbit without introducing any other elements into the expression. Another corollary is that in the case of a body moving in a parabolic orbit the velocity at any moment is that which would be acquired by the body in falling from an infinite distance to the place it occupies at the moment.
(4) If a number of bodies are projected from any point in space with the same velocity, but in various directions, and subjected only to the attraction of the sun, they will all return to the point of projection at the same moment, although the orbits in which they move may be ever so different.
(5) At each distance from the sun there is a certain velocity which a body would have if it moved in a circular orbit at that distance. If projected with this velocity in any direction the point of projection will be at the end of the minor axis of the orbit, because this is the only point of an ellipse of which the distance from the focus is equal to the semi-major axis of the curve, and therefore the only point at which the distance of the body from the sun is equal to its mean distance.
(6) The relation between the periodic time of a planet and its mean distance, approximately expressed by Kepler’s third law, follows very simply from the laws of centrifugal force. It is an elementary principle of mechanics that this force varies directly as the product of the distance of the moving body from the centre of motion into the square of its angular velocity. When bodies revolve at different distances around a centre, their velocities must be such that the centrifugal force of each shall be balanced by the attraction of the central mass, and therefore vary inversely as the square of the distance. If M is the central mass, n the angular velocity, and a the distance, the balance of the two forces is expressed by the equation
an² = M/a²,
whence a³n² = M, a constant.