Establish the given formula for a convex surface : μ 2 - μ 1 R = μ 2 v - μ 1 u

One medium (rarer) and on the right side the second medium (denser). P is the pole of the spherical surface and C is its centre of curvature. According to ray diagram of fig, I is the image of the object O. The normal drawn from point M on principal axis is MP’. According to Snell's law, μ 2 1 = sin i sin r where μ 2 1 is the refractive index of medium (2) with respect to medium (1). μ 2 1 = sin i sin r = μ If the angles i and r are very small, then μ = i r ⇒ i = μ r --------(1) In triangle, the exterior angle is equal to sum of the opposite two interior angles. Δ M O C i = α + γ -----------(2) Δ M I C , r = β + γ -----------(3) Substituting the value of i and r in equation (1), then we get, ( α + γ ) = n ( β + γ ) --------- (4) f point M is not far from the principal axis, then (i) Points P and P' are very close to each other and considered to be same point. (ii) Angles α,β and γ will be small. α ≈ tan α = M P O P = M P - u β = tan β = M P I P = M - v γ = tan γ = M P C P = M P R Now, substituting the values of α,β and γ in equation (4) and solving : μ v - 1 u = μ - 1 R Now again writing μ 2 1 in place of μ , we get μ 2 1 v - 1 u = μ 2 1 - 1 R On solving we get : μ 2 - μ 1 R = μ 2 v - μ 1 u