Let 'P' be the pole, 'C' be the centre of curvature and 'R' be the radius of curvature of a small aperture spherical refracting surface.
In triangle PMO: tan MOP = tan \(\alpha\) = \(\frac{PM}{PO}\)
In triangle PMI: tan MIP = tan \(\beta\) = \(\frac{PM}{PI}\)
In triangle MCP: tan MCP = \(tan \gamma\) = \(\frac{PM}{PC}\)
In triangle OMC, \(i = \alpha+\gamma \) = \(\frac{PM}{PO}+\frac{PM}{PC}\)
In triangle MCI, \(\gamma = r + \beta \)
r = \(\gamma - \beta = \frac{PM}{PC} - \frac{PM}{PI}\)
By snell's law,
\(n_1sin i = n_2sin r\)
\(n_1i=n_2r\)
\(n_1[\frac{PM}{PO}+\frac{PM}{PC}] = n_2[\frac{PM}{PC}-\frac{PM}{PI}]\)
\(\frac{n_1}{PO}+\frac{n_1}{PC}\) = \(\frac{n_2}{PC}-\frac{n_2}{PI}\)
-\(\frac{n_1}{u} + \frac{n_1}{R}\)= \(\frac{n_2}{R}-\frac{n_2}{v}\)
\(\therefore \frac{n_2}{v}-\frac{n_1}{u}=\frac{n_2-n_1}{R}\)
