Referring to fig. 55, we have, for obtaining the radius of the central disc, the following formulæ, in which _r_ = AB, half the aperture of the lens _r′_ = AB′ φ = AF, the focal distance _t′_ = A _a_, the thickness of the lens at the vertex _t″_ = B _b_, the thickness of the joint μ = the index of refraction ρ = the radius of curvature. Then for the radius of curvature near the axis we have: ( _t′_) ρ′ = (μ - 1)(φ + ----) ( μ ) and for that near the margin we have: r tan _i′_ = - φ sin _i′_ sin _e_ = -------- μ _r′_ = _r_ - _t″_ . tan _e_ _r′_ tan _i_ = ---- φ sin _i_ sin ε = ------- μ _r_ ρ″ = ---------√(μ² - 2 μ cos _e_ + 1) μ sin _e_ and, finally ρ′ + ρ″[57] ρ = ------- 2 [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. [Illustration: Fig. 56.] Then sin PCG PG μ = ------- = --; sin GPC CG and also sin QFG QG μ = ------- = -- sin QGF QF Now, as P approaches A, we have ultimately PG = AG, QG = BG, and QF = BF; Therefore, putting AG = θ and AC = ρ′ AG = θ BG θ - _t′_ μ = -- = ------; μ = -- = --------, CG = θ - ρ′ BF φ from which μ θ - μ ρ′ = θ; and μ φ = θ - _t′_ and eliminating θ, we have μ² φ + μ ρ′ = μ φ + _t′_, whence, as above, ( _t′_) ρ′ = (μ - 1)(φ + ----) ( μ ) But as this value of the radius of curvature, as already stated, is calculated for rays near the axis, it would produce a notable aberration for rays incident on the margin of the lens. In order, therefore, to avoid the effects of aberration as much as possible, a second radius of curvature must be calculated, so that rays incident on the margin of the lens may be refracted in a direction parallel to the axis. This second value of the radius is called ρ″ in the text, and is found as follows (referring to fig. 57): Let FB′ _b_ _x_ be the course of a ray refracted in the direction _b_ _x_ parallel to the axis A _x′_. This ray meets the surface AB in the point B′, whose position may be found approximately by tracing the path of the ray FB, on the supposition that the surface of the refracting medium is produced in the directions AB, _a′_ _b′_. [Illustration: Fig. 57.] Let C be the centre of curvature (see fig. 57) α = AC _b_ the angle of emergence