_l_ = BE the breadth of the zone = _r_₁ - _r_₂ ρ = the radius of curvature = _b_ C = _m_ C φ = focal distance AF _t_ = thickness of the joint B _b_ _t″_ = B _b_ μ = refractive index of the glass _i_₁ = BFA _i_₂ = EFA _r_₁ _r_₂ Then tan _i′_₁ = ----; tan _i′_₂ = ---- φ φ sin _i′_₁ sin _i′_₂ sin _e_₁ = ---------; sin _e_₂ = --------- _μ_ _μ_ _r′_₁ = _r_₁ - _t″_ sin _e_₁; _r′_₂ = _r_₂ - _t″_ sin _e_₂ _r′_₁ _r′_₂ tan _i_₁ = -----; tan _i_₂ = ----- φ φ sin _i_₁ sin _i_₂ sin ε = --------; sin ε′ = -------- μ μ μ sin ε sin α = ---------------------; √(μ² - 2 μ cos ε + 1) μ sin ε′ sin α′ = ---------------------; η = α′ - ε′ √(μ² - 2 μ cos ε′ + 1) 2 cos ε′ and lastly ρ = -------------------------------------- 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: [Illustration: 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 = _r_₁; AB′ = _r′_₁; C _b_ = ρ AE = _r_₂; AE′ = _r′_₂; B _b_ = _t″_; BE = _r_₁ - _r_₂ = _l_ ε = the first angle of refraction _b_ B′ _k_ η = the second angle of refraction B′ _b_ C ε′ = the first angle of refraction _e_ E _k′_ η′ = the second angle of refraction _e_ _m_ C α = the angle of emergence _b_ C _q_ α′ = the angle of emergence _m_ C _q_ _i′_₁ = BFA; _i′_₂ = EFA; _i_₁ = B′FA; _i_₂ = E′FA _e_₁ = B _b_ B′; _e_₂ = E _e_ E′. Proceeding exactly as in the case of the central lens we shall have BA _r_₁ EA _r_₂ tan _i′_₁ = -- = ----; tan _i′_₂ = -- = ---- AF φ AF φ sin _i′_₁ sin _i′_₂ sin _e_₁ = ---------; sin _e_₂ = --------- μ μ _r′_₁ = _r_₁ - _t″_ sin _e_₁; _r′_₂ = _r_₂ - _t″_ sin _e_₂ _r′_₁ _r′_₂ tan _i_₁ = -----; tan _i_₂ = ----- φ φ sin _i_₁ sin _i′_₂ sin ε = --------; sin ε′ = --------- μ μ μ sin ε μ sin ε′ sin α = ---------------------; and sin α′ = ---------------------- √(μ² - 2 μ cos ε - 1) √(μ² - 2 μ cos ε′ + 1)