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- 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 1m, that is, in this case 23, 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′=π2, 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 23.
- 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 lest than three times AE, the curve opens, a form something similar to that in Fig. 30.
