sin ε = sin i₁μ; sin ε′ = sin i₂μ
sin α = μ sin ε √μ² - 2 μ cos ε + 1;
sin α′ = μ sin ε′ √μ² - 2 μ cos ε′ + 1; η = α′ - ε′
and lastly ρ = 2 cos ε′ 2 cos {η + ¹⁄₂(α - α′)} sin ¹⁄₂ (α - α′)
which is Fresnel’s value of the radius of curvature.[58]
[58] The following steps will conduct us to this expression:
Fig. 60.
Let B b f E ([fig. 60]) represent the section of a zone by a plane passing through the axis of the lens AF, C the centre of curvature, F the radiant point, and FB′ b x, FE′m x′ the course of the extreme rays which are transmitted through the zone (and the latter of which passes from E′ to e through a portion of the zone or lens in contact with that under consideration). Then putting
AB = r₁; AB′ = r′₁; C b = ρ