Page:Optics.djvu/105



This is the equation to any point $AQ$ 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, $=∆$ and $∠ AQR$ must be the same for the refracted rays answering to $=θ$ and $∠ AqR=θ′$. We may therefore equate to nothing the differential of our equation with respect to $Am$ and $=x$, considering $mP$ and $=y$ as invariable. This gives us

We have, moreover, between $tan θ′=NR⁄NP=AR−MP⁄AM=∆tanθ−y⁄x$ and $xtanθ′−∆tanθ+y=0$ the equation

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

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

Let $θ$, $θ′$, (Fig. 105.) represent two rays inclined to each other at an infinitely small angle, incident obliquely on a curved surface at $x$, $y$; $xdθ′⁄cosθ′^{2}−∆dθ⁄cosθ^{2}=0$, $θ$ the refracted rays; $θ′$, $sinθ=m·sinθ′$, normals.