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86

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′.
  1. 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.

  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·sinφcosφ, or sinφ′sinφ=2·cosφ′cosφ, that is, m=2cosφ′cosφ;
    and if s=sinφ,
    m√1−s2=2√1−m2s2; ∴ s2=√4−m23m2=√727, if m=32.
    From this we find φ, or ERQ=30° 3612′, ERv=49° 48′.
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