A glass plate has a thickness ‘t’ and refractive index $\mathrm{\mu }$ . The angle of incidence of a ray from air into the plate is equal to the critical angle for glass-air intrerface. The lateral shift (perpendicular distance between incident ray and emergent ray) of ray is given by

Lateral shift is $\frac{\mathrm{tsin}(\mathrm{i}-\mathrm{r})}{\cos \mathrm{r}}$ where $\mathrm{i}$ and $\mathrm{r}$ are angle of incidence and refraction.

As the angle of incidence is the critical angle of glass and air interface

$\mathrm{sini}=\frac{1}{\mathrm{\mu }}\Rightarrow \mathrm{cosi}=\frac{\sqrt{{\mathrm{\mu }}^2-1}}{\mathrm{\mu }}$

At the surface of incidence, using Snell's law

$\mathrm{sini}=\mathrm{\mu sinr}$

$\Rightarrow \mathrm{sinr}=\frac{1}{{\mathrm{\mu }}^2}\Rightarrow \mathrm{cosr}=\frac{\sqrt{{\mathrm{\mu }}^4-1}}{{\mathrm{\mu }}^2}$

using this values in the formula for lateral shift, we get,

 lateral shift = $\frac{\mathrm{t}}{\mathrm{\mu }}\left(1-\frac{1}{\sqrt{{\mathrm{\mu }}^2+1}}\right)$