$4.$ The separated doublet, shown in figure, is formed of two thin doublets $($lenses $(1)$ and $(2),$ separated by distance $\backslash ($ d $\backslash ).$ Each one of the achromatized thin doublets consists of fused a planar$-$convex lens and a diverging lens made of BK$7$ and F$2$ glass, respectively. The focal length of lens $(1)$ is $3$ cm $,$ while that of lens $(2)$ is $5$ cm $.$

a$.$ Calculate the radii of the surfaces of the achromatized fused lenses composing the thin doublets $(1)$ and $(2).$

b$.$ If the separated doublet is used as an eyepiece lens having a focal length of $2\mathrm{.}5$ cm $,$ find the separation between the two thin doublets that will eliminate chromatic aberrations.

c$.$ Calculate the focal lengths of the eyepiece for the Yellow, Blue and Red Fraunhofer lines. How much does each focal length vary from the $2\mathrm{.}5$ cm desired value?