Page:Optics.djvu/111

 When $EQ=2⁄3AE$, (Fig. 120.) the segment of a circle just touches the sphere; all rays incident on $Am$ are refracted accurately, so as to meet in $q·(Aq=5⁄2AE)$. The other rays falling on $am$ form a caustic $mq′·(aq′=1⁄4aE)$.

From this last place of $Q$ to the centre, the caustic takes a form of the kind shown in Fig. 121., in which $EQ$ is half the radius, $q$ is on the surface $qq′=2⁄5qE$.

When $Q$ is at the center, of course there is no caustic other than that point itself: afterwards we find the figures described above, only that their places are inverted.

113. We have yet to consider the case of rays passing out of a denser into a rarer medium through a convex surface.

Let then $E$ (Fig. 122.) be the centre of a surface $CAc$ bounding a dense medium, and first let the incident rays be parallel to the axis $AE$.

The principal focus $F$ is at the distance $r⁄m−1$ from $A$, that is, at two radii if $m=3⁄2$: and if $AEm$ be the angle whose sine is $2⁄3$, $Qm$ will be the extreme ray that can be refracted, so that the caustic will touch the surface at $m$, and have a cusp at $F$.

As $Q$ comes towards $A$, (Fig. 123.) the caustic contracts both in length and breadth, till on $Q$ coinciding with $A$, it is reduced to that single point.

114. With respect to the forms of caustics belonging to curved surfaces not spherical, it is not worth while to say much, as the subject is difficult and of very little importance. There is, however, one case which is simple enough, namely, that in which the section of the refracting surface is a logarithmic or equiangular spiral, and the incident rays meet in its pole.

Let $Q$ (Fig. 124.) be the pole of such a spiral, $QR$, $QR′$, $QR″$ three successive rays refracted into the directions $RS$, $R′S′$, $R″S″$, so as to meet in $q$, $q′$.

Then, since in this case the angle of incidence is every where the