Let SPS' be a convex mirror surface separating a rarer medium of absolute refractive index n₁ from a denser medium of refractive index n₂.Let P be the centre of curvature of the surface .PC is the radius of curvature R of the surface.Let a point-object O placed on the principal axis PC produced backwards.An incident ray of light OA,after reflection at the point A on the surface,bends towards the normal CAN and goes along AI in the denser medium.Another incident ray OP falls on the refracting surface normally and goes un-deviated into the denser medium.The 2 refracted rays meet at the principal axis at I which is the real image of O.
Suppose the angle of incidence OAM=i,the angle of refraction IAC=r ,Po=-u,PI=+v and PC=+R.According to the coordinate geometry sign convention,u is -ve while v and R is +ve.Let α,β,γ b the angle made by OA,IA,Ca respect. with proncipal axis.Let h be the height of the normal AM dropped from A on the principal axis.Now,in ΔCOA,
by snell's law:
or n₁ sin i= n₂ sin r
or n₁i=n₂r......................[since the spherical mirror is of small aperture]
therefore the required eq. is