Page:Optics.djvu/48

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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

$p 2 = a 2 b 2 / a 2 + b 2 - u 2$ ,

we have

$p 2 / u = a 2 b 2 / u (a 2 + b 2 - u 2)$ ,

and $dl p 2 / u / du = 1 / u + 2 u / a 2 + b 2 - u 2 = 3 u 2 -(a 2 + b 2) / u (a 2 + b 2 - u 2)$ ;

$∴ v = a 2 + b 2 - u 2 / 3 u 2 -(a 2 + b 2) u$ .

Hence, when $u = a$ ,	$v = b 2 / 2 u 2 - b 2 a$ ,
and when $u = b$ ,	$v = a 2 / 2 b 2 - a 2 b$ .

The former of these values is always essentially positive, since $a$ is supposed to represent the semi-axis major, and therefore $2 a 2$ must be greater than $b 2$ ; but $2 b 2$ may be equal to, or greater than $a 2$ , so that when $u = b$ , $v$ may be infinite or negative.

When $2 b 2 > a 2$ , the form of the caustic is such as that shewn in Fig. 22.

When $b = \sqrt 3 / 2 a$ , $u = b$ gives $v =2 b$ , and the curve is that of Fig. 23.

When $2 b 2 = a 2$ , we have infinite branches asymptotic to the axis minor, as in Fig. 24.

When $2 b 2 < a 2$ , 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.