Three thin prisms are combined as shown in the figure. The...

The angular dispersion produced by a single thin prism of refracting angle $A$ is given by $\theta = (\mu_v - \mu_r)A$.

The given combination consists of two identical crown glass prisms (each with angle $A$) and one flint glass prism (with angle $A'$). The flint glass prism is placed inverted relative to the crown glass prisms.

The total angular dispersion of the combination is the sum of the dispersions of the individual prisms. Since the flint prism is inverted, its dispersion will be in the opposite direction:

$\theta_{\text{net}} = \theta_{\text{crown}} + \theta_{\text{flint}} + \theta_{\text{crown}}$

$\theta_{\text{net}} = (\mu_v - \mu_r)A - (\mu_v' - \mu_r')A' + (\mu_v - \mu_r)A$

$\theta_{\text{net}} = 2(\mu_v - \mu_r)A - (\mu_v' - \mu_r')A'$

For no net angular dispersion, $\theta_{\text{net}} = 0$:

$2(\mu_v - \mu_r)A - (\mu_v' - \mu_r')A' = 0$

$(\mu_v' - \mu_r')A' = 2(\mu_v - \mu_r)A$

$\dfrac{A'}{A} = \dfrac{2(\mu_v - \mu_r)}{\mu_v' - \mu_r'}$