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

$p2=.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac.nobar .num{display:block;line-height:1em;margin:0.0em 0.1em}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}⁠a2b2/a2+b2−u2⁠$ ,

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.