Derive the relation between object distance $(u)$,image distance $(v)$,refractive indices of the media ($n_1$ and $n_2$),and the radius of curvature $(R)$ for a spherical refracting surface.

(N/A) Consider a spherical surface separating two media of refractive indices $n_1$ and $n_2$. Let $O$ be the object and $I$ be the image formed on the principal axis.
For small angles,we have:
$\alpha = \angle NOM \approx \frac{MN}{OM} = \frac{MN}{-u}$
$\beta = \angle NCM \approx \frac{MN}{MC} = \frac{MN}{R}$
$\gamma = \angle NIM \approx \frac{MN}{MI} = \frac{MN}{v}$
From the geometry of the triangles:
In $\Delta NOC$,$i = \alpha + \beta = \frac{MN}{-u} + \frac{MN}{R}$
In $\Delta NIC$,$\beta = r + \gamma \implies r = \beta - \gamma = \frac{MN}{R} - \frac{MN}{v}$
Applying Snell's Law for small angles,$n_1 i = n_2 r$:
$n_1 \left( \frac{MN}{-u} + \frac{MN}{R} \right) = n_2 \left( \frac{MN}{R} - \frac{MN}{v} \right)$
Dividing by $MN$ and rearranging:
$\frac{n_2}{v} - \frac{n_1}{u} = \frac{n_2 - n_1}{R}$