(11)
In the case of an ellipse described about the centre as pole we have
| a2b2 | = a2 + b2 − r2; |
| p2 |
(12)
hence P = μr, if μ = h2/a2b2. This merely shows that a particular ellipse may be described under the law of the direct distance provided the circumstances of projection be suitably adjusted. But since an ellipse can always be constructed with a given centre so as to touch a given line at a given point, and to have a given value of ab (= h/√μ) we infer that the orbit will be elliptic whatever the initial circumstances. Also the period is 2πab/h = 2π/√μ, as previously found.
| Fig. 68. |
Again, in the equiangular spiral we have p = r sinα, and therefore P = μ/r3, if μ = h2/sin2 α. But since an equiangular spiral having a given pole is completely determined by a given point and a given tangent, this type of orbit is not a general one for the law of the inverse cube. In order that the spiral may be described it is necessary that the velocity of projection should be adjusted to make h = √μ·sinα. Similarly, in the case of a circle with the pole on the circumference we have p2 = r2/2a, P = μ/r5, if μ = 8h2a2; but this orbit is not a general one for the law of the inverse fifth power.
In astronomical and other investigations relating to central forces it is often convenient to use polar co-ordinates with the centre of force as pole. Let P, Q be the positions of a moving point at times t, t + δt, and write OP = r, OQ = r + δr, ∠POQ = δθ, O being any fixed origin. If u, v be the component velocities at P along and perpendicular to OP (in the direction of θ increasing), we have
| u = lim. | δr | = | dr | , v = lim. | r δθ | = r | dθ | . |
| δt | dt | δt | dt |
(13)