Page:Optics.djvu/105

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81

Let⁠ $AQ$	$=∆$ ,
$∠ AQR$	$= θ$ , $∠ AqR = θ'$ ,
$Am$	$= x$ ,
$mP$	$= y$ .

Then $tan θ' = NR / NP = AR - MP / AM = ∆tan θ - y / x$ ,

$x tan θ' -∆tan θ + y =0$ .

This is the equation to any point $P$ on the refracted ray. If this point be on the caustic, it must be common to two successive refracted rays infinitely near each other, that is, $x$ and $y$ must be the same for the refracted rays answering to $θ$ and $θ + dθ$ . We may therefore equate to nothing the differential of our equation with respect to $θ$ and $θ'$ , considering $x$ and $y$ as invariable. This gives us

$x dθ' / cos θ' 2 -∆ dθ / cos θ 2 =0$ .

We have, moreover, between $θ$ and $θ'$ the equation

$sin θ = m \cdotsin θ'$ .

These three equations must, by the elimination of the functions of $θ$ and $θ'$ , give the one containing only $x$ , $y$ , and $∆$ , which will be the equation to the caustic.

107. Prop. Required the focus of a thin pencil of rays after being refracted obliquely at a curved surface.

Let $QR$ , $QR'$ , (Fig. 105.) represent two rays inclined to each other at an infinitely small angle, incident obliquely on a curved surface at $R$ , $R'$ ; $Rq$ , $R'q$ the refracted rays; $RE$ , $R'E$ , normals.

Let⁠ $EQ$	$= q$ ,
$Eq$	$= t$ ,
$ER$	$= r$ ,
$QR$	$= u$ ,
$Rq$	$= v$ ,
$∠ QRZ$	$= φ$ ,
$∠ ERq$	$= φ'$ .

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