Then for the radius of curvature near the axis we have:
ρ′ = (μ - 1)(φ + t′μ)
and for that near the margin we have:
tan i′ = rφ
sin e = sin i′μ
r′ = r - t″ . tan e
tan i = r′φ
sin ε = sin iμ
ρ″ = rμ sin e √μ² - 2 μ cos e + 1
and, finally ρ = ρ′ + ρ″ 2[57]
[57] The following steps lead to the formulæ given in the text. Let APQB ([fig. 56]) represent a section of the central lens by a plane passing through its axis AF; F the focus for incident rays; and FQPH the path of a ray refracted finally in the direction PH, parallel to the axis. Let C be the centre of curvature, then PC is a normal to the curve at P; and, producing PQ to meet the axis in G, we have G the focus of the rays, after refraction at the surface BQ.
Fig. 56.
Then μ = sin PCGsin GPC = PGCG; and also μ = sin QFGsin QGF = QGQF