For a spherical surface,when refraction takes place from a rarer medium `(mu_1)` to a denser medium `(mu_2)`, we write
`-(mu_1)/u` + `(mu_2)/v` = `( mu_2-mu_1)/R`
where the symbols have their usual meanings. Now suppose that the object is placed in the denser to the rarer medium . Then, rewrite the above equation.

Explanation

Let O be the point object on the principal axis lying in denser medium.

A ray of light incident normally on the refracting surface along the principal axis, passes undeviated.
Another ray of light OA incident at point A after refraction bends away from the normal and on producing backwards meets the principal axis at I.
I is the virtual image of the object O.
Let i,r be the angle of incidence and angle of refraction.
Let `alpha, beta` and `gamma` be the angles made by the incidencet ray, refracted ray and normal respectively with the principal axis. Draw AN perpendicular on the principal axis.
From `DeltaAOC" "alpha=i+gamma" " or" "i=alpha-gamma` ........i
Sicne angles `alpha` and `gamma` are small (assumption) so they can be replaced by their tangents. Hence eqn (i) can be writen as
`i=tan alpha-tan gamma`...............ii
From `DeltaAIC" "r+gamma=beta` or `r=beta-gamma`..............iii
Since angles `beta` and `gamma` are small ( assumption) so they can be replaced by their tangents. Hence eqn (i) can be write
`r=tan beta-tan gamma`.........iv
Determine of `tan alpha, tan beta` and `tan gamma`
From rt. `/_d DeltaANO" "tan alpha=(AN)/(NO)`........v
From rt. `/_dDeltaANI," "tan beta=(AN)/(NI)`..........vi
From rt.`/_d DeltaANC" "tan gamma=(AN)/(NC)`.......vii
using eqns (v) to (vii) in eqns (ii) and (iv) we get
`i=(AN)/(NO)-(AN)/(NC)`.........viii
and `r=(AN)/(NC)-(AN)/(NI)`............ix
According to Snell.s law
`(sini)/(sinr)=(mu_(1))/(mu_(2))` or `mu_(2)sini=mu_(2)sinr`......x
Since angles i and r are small so sin i=i and sinr=r. Hence eqn x becomes
`mu_(2)i=mu_(1)r`....xi
using eqn viii and ix in eqn xi we get
`mu_(2)[(AN)/(NO)-(AN)/(NC)]=mu_(1)[(AN)/(NI)-(AN)/(NC)]` or `(mu_(2))/(NO)-(mu_(2))/(NC)-(mu_(1))/(NI)-(mu_(1))/(NC)`..........xii
Since aperture of the spherical surface is small, so point Nlies very close to pint P.
`:.NO~=PO,NC~=PC` and `NI~=PI`
Hence eqn xii becomes
`(mu_(2))/(PO)-(mu_(2))/(PC)=(mu_(1))/(PI)-(mu_(1))/(PC)`..............xiii
Applying new cartesian sign conventions
`PO=-u,PC=-R` and `PI=-v`
`:.` Eqn xiii can be written as
`(mu_(2))/(-u)+(mu_(2))/R=-(mu_(1))/v+(mu_(1))/R` or `-(mu_(2))/u+(mu_(1))/v=-(mu_(2))/R+(mu_(1))/r`
or `-(mu_(2))/u+(mu_(1))/v=(mu_(1)-mu_(2))/R`.xiv