Page:Optics.djvu/110

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3. For instance, when Q is at the extremity of the diameter of a sphere, (Fig. 118.)

   AQ=	2AE,
   Aq=	4AE,
   EmQ=	41° 49′,
   QEm=	96° 22′;

   Rv, R′v are the asymptotes;^[1]

   Ev	=3.949AE,
   EvR	=11° 25′.

4. Let now Q come within the sphere, (Fig. 119.).
   Provided EQ be greater than 2/3AE, a segment of a circle on EQ capable of containing an angle of 41° 49′, will cut the section of the sphere in two points m, n, at which rays incident from Q will be refracted parallel to the surface. Between the points m, n, there will be no refraction: those rays which fall on Am will, after refraction, form a caustic of the same kind as that of the last case: those which fall on an will form another caustic nq′, q′ being the focus for rays refracted at α.

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1. The place of the asymptote is thus calculated:
   
   Since v is to be infinite, and u=2rcosφ,

   2rcosφ=rcosφ·tanφ′/tanφ′−tanφ.

   Hence,

   sinφ′/cosφ′=2·sinφ/cosφ, or sinφ′/sinφ=2·cosφ′/cosφ, that is, m=2cosφ′/cosφ;

   and if s=sinφ,

   m√1−s²=2√1−m²s²; ∴ s²=√4−m²/3m²=√7/27, if m=3/2.

   From this we find φ, or ERQ=30° 361/2′, ERv=49° 48′.