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1. whose sine is 2/3. Let $ECD$ be this angle. Applying the construction discovered above, we find that if $Em$ be drawn perpendicular to $CD,\ m$ is at the extremity or edge of the caustic, which must be of a form something like $mnFn'm$.
   If the refracting surface be only part of a hemisphere, as GAg^{[errata 1]}, the caustic is of course reduced to $nFn',$ if $n,\ n'$ be the points where the rays refracted from $G,\ g$ meet the whole caustic.
2. As $Q$ advances towards the surface, (Fig. 108.) the caustic diminishes in breadth, and increases in length, till when $AQ=2AE,\ AQ$ becomes infinite, and then the caustic has two infinite branches asymptotic to the axis, (Fig. 109.).
3. Past that point, $q$ comes on the same side with $Q$, and the caustic breaks as it were into two parts, (Fig. 110.) one of which proceeding from $q$ is imaginary, the other is real, and both have infinite branches extending along the asymptotes $Oo,\ Oo'.$
4. When $Q$ comes to $A$, both parts of the caustic, of course, disappear altogether, being entirely merged in that point.

110. When the refracting surface is a concave hemisphere, the caustic lies on the same side with the radiant point.

1. If the incident rays be parallel to the axis, the form of the caustic is that represented in Fig. 111, where $F$ is the principal focus, $QC$ the extreme incident ray, $Cm$ the extreme refracted ray touching the caustic at its lip $m,\ Em$, is perpendicular to $Cm$, and the point $m$ is found by drawing $Cm$ at the proper angle, (that whose sine is 1/m) intersecting a semi-circle on $CE$.
2. As $Q$ advances towards $E$, the caustic contracts (Fig. 112.) till when $AQ=(m+1).AE,$, as there is no aberration, it vanishes altogether.
3. It then turns its arms the contrary way, as in Fig. 113, the rays refracted on one side of the axis intersecting on the other, till $Q$ gets to the centre, when the caustic again vanishes.

Errata

1. Original: Gag was amended to GAg: detail