Note 12 — Chapter IV · THE PROPERTIES OF OBJECTIVES AND MIRRORS

The curvature of the image is the thing which sets a limit to shortening the relative focus, as already noted, for the astigmatic image surfaces as we have seen, fall rapidly apart away from the axis, and both curvatures are considerable. The tangential is the greater, corresponding roughly to a radius notably less than ⅓ the focal length, while the radial fits a radius of less than ⅔ this length with all ordinary glasses, given forms correcting the ordinary aberrations. The curves are concave towards the objective except in “anastigmats” and some objectives having bad aberrations otherwise. Their approximate curvatures assuming a semiangular aperture for an achromatic objective not over say 5°, have been shown to be, to focus unity

ρ_r = 1 + (1/(ν-ν′)(ν/n - ν′/n′), and ρ_t = 3 + 1/(ν-ν′)(ν/n - ν′/n′)

ρ_r and ρ_t being the respective reciprocals of the radii. The surfaces are really somewhat egg shaped rather than spherical as one departs from the axis.