(16)
where P, as before, denotes the acceleration towards O. If in this we put r = 1/u, and eliminate t by means of (15), we obtain the general differential equation of central orbits, viz.
| d2u | + u = | P | . |
| dθ2 | h2u2 |
(17)
If, for example, the law be that of the inverse square, we have P = μu2, and the solution is of the form
| u = | μ | {1 + e cos (θ − α)}, |
| h2 |
(18)
where e, α are arbitrary constants. This is recognized as the polar equation of a conic referred to the focus, the half latus-rectum being h2/μ.
The law of the inverse cube P = μu3 is interesting by way of contrast. The orbits may be divided into two classes according as h2 ≷ μ, i.e. according as the transverse velocity (hu) is greater or less than the velocity √μ·u appropriate to a circular orbit at the same distance. In the former case the equation (17) takes the form