24
28. Required the form of the caustic when the reflecting curve is an ellipse, and the radiating point its centre. Fig. 22, &c.
The polar equation to an ellipse about its centre being
p2=a2b2a2+b2−u2,
we have
p2u=a2b2u(a2+b2−u2),
and dlp2udu=1u+2ua2+b2−u2=3u2−(a2+b2)u(a2+b2−u2);
∴ v=a2+b2−u23u2−(a2+b2)u.
| Hence, when u=a, | v=b22u2−b2a, |
| and when u=b, | v=a22b2−a2b. |
The former of these values is always essentially positive, since a is supposed to represent the semi-axis major, and therefore 2a2 must be greater than b2; but 2b2 may be equal to, or greater than a2, so that when u=b, v may be infinite or negative.
When 2b2>a2, the form of the caustic is such as that shewn in Fig. 22.
When b=√32a, u=b gives v=2b, and the curve is that of Fig. 23.
When 2b2=a2, we have infinite branches asymptotic to the axis minor, as in Fig. 24.
When 2b2<a2, there are asymptotes inclined to that line (Fig. 25.).
29. There are some simple cases in which it is easy to determine the nature of the caustic by geometrical investigation.
