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 third of $AQ$. Then, since the angle $ORp$ is double of $OPp$, it must be equal to the angle $AEP$, which is double of $EPQ$, and the radii $EO$, $RO$ being equal, it follows, that the arcs $Op$, $OV$, must in every case be equal, and that the locus of the point $p$ is an epicycloid described by the revolution of a circle equal to $PpO$ round that of which $OV$ is a part.

30. These are the only cases in which the caustic to a spherical reflector is a known curve; we may, however, without much difficulty, make out the kind of figure that it assumes in other cases, though the investigation of the equation is very tedious.

When the radiant point is at a finite distance without the circle, the form of the caustic is naturally intermediate to those already found. It is represented in Fig. 27. The caustic touches the circle at the points $C$, $c$ where tangents from $Q$ meet it, and it has a cusp at $V$, the focus of the principal reflected rays. There is also an imaginary branch $Cvc$.

When $Q$ is within the circle, and $EQ$ is greater than half the radius, the caustic takes the form represented in Fig. 30, with a cusp at $V$, and two others at $C$, $c$. There are two infinite branches extending along the asymptotes $DG$, $DG′$.

There is another part with a cusp at $v$, and two infinite branches having the same lines $DG$, $DG′$ for asymptotes.

When $Q$ bisects the radius $EB$, the caustic is such as represented in Fig. 30. $V$ bisects $EF$, and the axis $EA$ supplies the place of the asymptotes of the last case.

When $EQ$ becomes less than half the radius, the caustic contracts to the form shown in Fig. 31. $EV$ is then less than half $EF$.

When $Q$ comes to the centre, the caustic is reduced to that one point.

31. Supposing the form of the reflecting surface to be that generated by the revolution of a cycloid about its axis, and the incident rays to be parallel to that axis, it is required to describe the caustic. (Fig. 32.)

Let $Pp$ be a reflected ray, touching the caustic in $p$, $PG$ the normal. Then, since $Pp$ must be one-fourth of the chord of the