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oscillating circle, and that in the cycloid, the normal is one-fourth of the diameter of that circle, $GpP$ must be a right angle. Let $GK$ be the diameter of the generating circle of the cycloid, when the describing point is at $P$; $H$ the intersection of this with $Pp$. Then, since $HpG$ has been proved to be a right angle, a circle passing through $H$, $p$, $G$, will have $HG$ for its diameter. Moreover $H$ is the centre of the circle $GPK$, for the angle $HGP$ is equal to $GPN$, and consequently to $GPH$. Since then the radius of the smaller circle $HpG$ is half that of the other, and since the angle GHp^{[errata 1]} is double of the angle $GPr$, it follows, that the arcs $Gp$, $Gr$ are equal, that $Gp$ is equal to $GD$, and that the locus of $p$ is a cycloid having $GpH$ for its generating circle.

The surface generated by the revolution of this cycloid about $AD$, has of course a cusp at $D$.

32. Enough has probably been said on this subject, which might without much difficulty be prosecuted further, but it is one rather of curiosity than utility, and more suited to the speculative Geometer, or Algebraist, than the practical Optician.

33. Some writers have investigated expressions for the density of rays, or brightness, at different parts of a caustic: it may be sufficient to observe, that it is much greatest at and near a cusp.

34. Some of the forms of caustics above described may be exhibited in an imperfect manner. If a concave reflector^[1] be placed so as to reflect directly the rays of the Sun admitted through an aperture into a dark room in which there is a good deal of dust or smoke, there will be observed a bright funnel-shaped form, such as that represented in Fig. 33. This, however, is not a simple caustic, such as that described in Fig. 26, because the solar rays would not in any case converge to one single point, but to a circular image, as we shall see in the next Chapter.

35. It is somewhat remarkable, that an infinite number of different mirrors may reflect rays proceeding from a given point, so as to produce the same caustic.

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1. The inside of the cover of a watch will answer the purpose extremely well.

Errata

1. Original: GHP was amended to GHp: detail