Page:Optics.djvu/109

 #While $Q$ is between $E$ and $A$, we have a similar kind of curve, and when $Q$ is at $A$, the caustic consists of a curve with two branches very much bent back, and of the point $A$ itself, (Fig. 114.). 111. Let now the refraction take place out of the denser into the rarer medium, and as a first instance, let the refracting body be a glass hemisphere, and the rays enter perpendicularly at the flat surface, (Fig. 115.).

In the first place we may remark, that as no refraction can take place at an angle of incidence greater than that whose sine is $1⁄m$, that is, in this case $2⁄3$, if $En$ be taken two-thirds of $EC$, and $nm$ be drawn parallel to $EA$, $Qm$ will be the extreme ray that can be refracted.

Since $v=0$, when $φ′=π⁄2$, or $cosφ′=0$, it is plain that the curve must begin at $m$, $m′$ touching the circle, and extend to $F$, the principal focus.

As to the rays that are without the limits of refraction, they are of course reflected at the concave surface, and their caustic consists of parts of two epicycloids, $CV$, $cv$.

A plano-convex lens represented by $mAm′$ would give the whole of the caustic $mFm′$.

112. Let now the radiant point be in the axis of a cylinder of glass terminated by a convex hemisphere, (Fig. 116.) Suppose $AQ>3·AE$.The caustic here extends further both in length and breadth than in the last case. It begins of course at the point $m$, $EmQ$ being the angle whose sine is $2⁄3$. When $AQ=3AE$, $Aq$ is infinite, so that the branches of the caustic become asymptotic to the axis, as in Fig. 117. When $AQ$ is lestless [sic] than three times $AE$, the curve opens, a form something similar to that in Fig. 30.