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 =\\displaystyle\\frac{sin i}{sin r}\\\\) ........(1)but here i and r are the small then we put.\\\\(sin i=i\\\\) \\\\(sin r=r\\\\) in equation (1)\\\\(\\mu =\\displaystyle\\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\\\\(=\\displaystyle\\frac{arc}{radius}\\\\)\\\\(\\alpha =\\displaystyle\\frac{PA}{OP}\\\\)\\\\(\\beta =\\frac{PA}{IP}\\\\)now \\\\(\\gamma =\\displaystyle\\frac{PA}{CP}\\\\)using value of \\\\(\\alpha, \\beta, \\gamma\\\\) in equation (5)\\\\(\\displaystyle\\frac{PA}{PC}-\\frac{PA}{PO}=\\mu\\left(\\displaystyle\\frac{PA}{PC}-\\frac{PA}{PI}\\right)\\\\)\\\\(PA\\left(\\displaystyle\\frac{1}{PC}-\\frac{1}{PO}\\right)=m. PA\\left(\\displaystyle\\frac{1}{PC}-\\frac{1}{PO}\\right)\\\\)\\\\(\\displaystyle\\frac{1}{PC}-\\frac{1}{PO}=\\mu\\left(\\displaystyle\\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(\\displaystyle\\frac{1}{-R}\\right)-\\left(\\displaystyle\\frac{1}{-u}\\right)=\\mu \\left(\\displaystyle\\frac{-1}{R}+\\frac{1}{V}\\right)\\\\)\\\\(\\displaystyle\\frac{-1}{R}+\\frac{1}{u}=\\mu\\left(\\displaystyle\\frac{-1}{R}+\\frac{1}{v}\\right)\\\\)\\\\(\\displaystyle\\frac{-1}{R}+\\frac{1}{u}=\\frac{\\mu}{R}+\\frac{\\mu}{v}\\\\)\\\\(\\displaystyle\\frac{-1}{R}+\\frac{\\mu}{R}=\\frac{\\mu}{v}-\\frac{1}{u}\\\\)\\\\(\\displaystyle\\frac{\\mu -1}{R}=\\frac{\\mu}{v}-\\frac{1}{u}\\\\)This is the required expression for the refraction formula for the concave spherical surface.