1. Given data

• Both prisms are equilateral, so prism angle $A=60^\circ.$
• Refractive indices: $\mu_1=\sqrt{\frac32},\qquad \mu_2=\sqrt3.$
• The ray emerging from prism $P_1$ enters prism $P_2$.
• It is given that in prism $P_2$, the ray undergoes minimum deviation.

2. Condition for minimum deviation in $P_2$

For a prism at minimum deviation, $r_1=r_2=\frac{A}{2}=30^\circ.$

So, inside prism $P_2$, the angle of refraction at the first face must be $30^\circ$.

Applying Snell's law at the first face of $P_2$ (vacuum to prism): $\sin i_2=\mu_2\sin 30^\circ$ where $i_2$ is the angle of incidence on prism $P_2$ from vacuum.

Thus, $\sin i_2=\sqrt3\cdot \frac12=\frac{\sqrt3}{2}$ so $i_2=60^\circ.$

Hence, the ray must strike the first face of $P_2$ at angle $60^\circ$ to the normal.

3. Relating this to the ray emerging from $P_1$

Since the corresponding faces of the two prisms are parallel, the normal to the second face of $P_1$ is parallel to the normal to the first face of $P_2$.

Therefore, the angle of emergence from $P_1$ into vacuum must be $e_1=i_2=60^\circ.$

4. Refraction at the second face of $P_1$

Let the internal angle of incidence at the second face of $P_1$ be $r_2$.

Using Snell's law from prism $P_1$ to vacuum: $\mu_1\sin r_2=\sin e_1$ $\sqrt{\frac32}\sin r_2=\sin 60^\circ=\frac{\sqrt3}{2}.$

So, $\sin r_2=\frac{\sqrt3/2}{\sqrt{3/2}}=\frac{1}{\sqrt2}.$ Hence, $r_2=45^\circ.$

5. Use prism relation for $P_1$

For prism $P_1$, $r_1+r_2=A=60^\circ.$ Thus, $r_1=60^\circ-45^\circ=15^\circ.$

6. Refraction at the first face of $P_1$

Let the external angle of incidence be $\theta$.

By Snell's law (vacuum to prism $P_1$): $\sin\theta=\mu_1\sin r_1$ $\sin\theta=\sqrt{\frac32}\sin 15^\circ.$

So, $\theta=\sin^{-1}\left[\sqrt{\frac32}\sin 15^\circ\right].$

This is of the form $\theta=\sin^{-1}\left[\sqrt{\frac32}\sin\left(\frac{\pi}{\beta}\right)\right].$

Therefore, $\frac{\pi}{\beta}=15^\circ=\frac{\pi}{12}$ which gives $\beta=12.$

7. Final answer

$\boxed{12}$

The derived answer matches the stored correct answer.