η = B′_b_ C the second angle of refraction ε = B b B′ the first angle of refraction _i_ = B′FA the first angle of incidence _i′_ = BFA _e_ = _b′_ B _b_ AB = _r_ AB′ = _r′_ B _b_ = _t″_ the thickness of the lens at the edge AF = φ the focal distance. Then _r_ sin _i′_ tan _i′_ = ---; sin _e_ = -------- φ μ whence _b_ _b′_ = _t″_ tan _e_ becomes known. Now, since BB′ = _b_ _b′_ nearly, AB′ = AB - _b_ _b′_ or _r′_ = _r_ - _t″_ tan _e_. From this is obtained the angle of incidence _i_, and the first angle of refraction ε; for _r′_ sin _i_ tan _i_ = ---- and sin ε = -------. φ μ Next B′ _b_ C = B _b_ C - B _b_ B′ or η = α - ε and sin α = μ sin η = μ sin (α - ε) sin α from which, sin α cos ε - cos α sin ε = ----- μ ( 1) whence sin α (cos ε - -) = cos α sin ε; and ( μ) ( 2 cos ε 1) sin² α (cos² ε - ------- + --) = cos² α sin² ε = ( μ μ²) (1 - sin² α) sin² ε = sin² ε - sin² α sin² ε Then transposing we have { 2 cos ε 1} sin² α {(cos² ε + sin² ε) - ------- + --} = sin² ε { μ μ²} and because (cos² ε + sin² ε) = 1 we have, by dividing, sin² ε μ² sin² ε sin² α = ------------------ = ------------------ { 2 cos ε 1} μ² - 2 μ cos ε + 1 {1 - ------- + --} { μ μ²} and μ sin ε sin α = --------------------- √(1 - 2 μ cos ε + μ²) Next, since _a′_ _b_ _r_ _b_ C = ---------- = -----, sin AC _b_ sin α putting C _b_ = _ρ″_, and substituting we have _r_ _ρ″_ = ------- √(μ² - 2 μ cos ε + 1) μ sin ε and, taking for the radius of curvature, the mean of _ρ′_ and _ρ″_ the values calculated for the central and marginal rays, we have finally ρ′ + ρ″ ρ = ------- 2 [Illustration: Fig. 58.] I come next to the _second_ case, which concerns the calculation of the elements of a concentric ring. The section _a_ _b_ _c_ _d_ _e_ (fig. 58) of one of those rings includes a mixtilinear triangle _a_ _b_ _e_, and a rectangle _b_ _c_ _e_ _d_, the thickness _b_ _c_ being the same as that of the edge of the central disc; and the elements to be determined are the radius of the curve surface, and the position of the centre of curvature, with reference to the vertex of the lens. [Illustration: Fig. 59.] The radius of curvature of the zone may be calculated by the following formulæ, in which (see fig. 59) _r_₁ = AB the distance of the outer margin of the zone from the axis of the lens _r_₂ = AE the distance of the inner margin from the axis