Now, as P approaches A, we have ultimately PG = AG, QG = BG, and QF = BF;
Therefore, putting AG = θ and AC = ρ′
μ = AGCG = θθ - ρ′; μ = BGBF = θ - t′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, ρ′ = (μ - 1)(φ + t′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′.
Fig. 57.
Let C be the centre of curvature (see [fig. 57])
α = AC b the angle of emergence