Derive the following expression for the refraction at concave spherical surface: $\frac{\mu }{v}-\frac{1}{u}=\frac{\mu -1}{R}$.

Let MPN be concabe spherical surface of a medium of refractive index $\mu$. P is the pole o is the centre of curvature PC is the principle axis. Let O be a point object or it's principle axis kept in a rarer medium first incident ray travelling through C is normal to the spherical surface therefor it will not deviate and goes along PX another mident ray OA will refract at Point A and it bends towards the normal. AB and PX are proceeded behind then at I, a virtual image will be form.
Let $\alpha ,\beta ,\gamma$ are the ray angle normal with the principle axis respectively.
then by snell's law
$\mu =\frac{sini}{sinr}$            ........(1)
but here i and r are the small then we put.
$sini=i$       $sinr=r$ in equation (1)
$\mu =\frac{i}{r}$
$i=\mu n$                 ...........(2)
now by using exterior angle theorem in $\Delta$AOC
$\gamma =1+a$
$i=\gamma -\alpha$             .........(3)
now in $\Delta$IAC by using exterior angle theorem
$\gamma =b+r$
$rho=\gamma -\beta$            ..........(4)
putting value of i and r in equation (2)
$(\gamma -\alpha )=\mu (\gamma -\beta )$           ...........(5)
now angle$=\frac{arc}{radius}$
$\alpha =\frac{PA}{OP}$
$\beta =\frac{PA}{IP}$
now $\gamma =\frac{PA}{CP}$
using value of $\alpha ,\beta ,\gamma$ in equation (5)
$\frac{PA}{PC}-\frac{PA}{PO}=\mu \left(\frac{PA}{PC}-\frac{PA}{PI}\right)$
$PA\left(\frac{1}{PC}-\frac{1}{PO}\right)=m.PA\left(\frac{1}{PC}-\frac{1}{PO}\right)$
$\frac{1}{PC}-\frac{1}{PO}=\mu \left(\frac{1}{PC}-\frac{1}{PI}\right)$         .........(6)
now by using sign convention
$PC=-R$                $PI=V$
$PO=-u$
using these value in equation (6)
$\left(\frac{1}{-R}\right)-\left(\frac{1}{-u}\right)=\mu \left(\frac{-1}{R}+\frac{1}{V}\right)$
$\frac{-1}{R}+\frac{1}{u}=\mu \left(\frac{-1}{R}+\frac{1}{v}\right)$
$\frac{-1}{R}+\frac{1}{u}=\frac{\mu }{R}+\frac{\mu }{v}$
$\frac{-1}{R}+\frac{\mu }{R}=\frac{\mu }{v}-\frac{1}{u}$

$\frac{\mu -1}{R}=\frac{\mu }{v}-\frac{1}{u}$

This is the required expression for the refraction formula for the concave spherical surface.