Page:Optics.djvu/108

 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 $$m n F n' m$$. If the refracting surface be only part of a hemisphere, as $$, the caustic is of course reduced to $$nFn',$$ if $$n,\ n'$$ be the points where the rays refracted from $$G, \ g$$ meet the whole caustic. As $$Q$$ advances towards the surface, (Fig. 108.) the caustic diminishes in breadth, and increases in length, till when $$AQ=2AE, \ AQ$$ becomes infinite, and then the caustic has two infinite branches asymptotic to the axis, (Fig. 109.). Past that point, $$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'.$$ When $$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.

Errata
 * 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.