Page:Optics.djvu/48

 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

we have

The former of these values is always essentially positive, since $p^{2}=a^{2}b^{2}⁄a^{2}+b^{2}−u^{2}$ is supposed to represent the semi-axis major, and therefore $p^{2}⁄u=a^{2}b^{2}⁄u(a^{2}+b^{2}−u^{2})$ must be greater than $dlp^{2}⁄u⁄du=1⁄u+2u⁄a^{2}+b^{2}−u^{2}=3u^{2}−(a^{2}+b^{2})⁄u(a^{2}+b^{2}−u^{2})$; but $∴ v=a^{2}+b^{2}−u^{2}⁄3u^{2}−(a^{2}+b^{2})u$ may be equal to, or greater than $u=a$, so that when $v=b^{2}⁄2u^{2}−b^{2}a$, $u=b$ may be infinite or negative.

When $v=a^{2}⁄2b^{2}−a^{2}b$, the form of the caustic is such as that shewn in Fig. 22.

When $a$, $2a^{2}$ gives $b^{2}$, and the curve is that of Fig. 23.

When $2b^{2}$, we have infinite branches asymptotic to the axis minor, as in Fig. 24.

When $a^{2}$, 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.