The dispersive powers of crown and flint glasses are `0.03 and 0.05` respectively. The difference in refractive indices for blue and red colour is `0.015` for crown glass and `0.022` for flint glass. Calculate the angles of the two prisms for a deviation of `2^@` without dispersion.

Explanation

To solve the problem step by step, we will use the given data and apply the relevant formulas.

### Given Data:
- Dispersive power of crown glass (ω) = 0.03
- Dispersive power of flint glass (ω') = 0.05
- Difference in refractive indices for crown glass (μB - μR) = 0.015
- Difference in refractive indices for flint glass (μB' - μR') = 0.022
- Total deviation (D) = 2°

### Step 1: Write the formula for deviation without dispersion
The formula for the total deviation (D) for two prisms is given by:
\[ D = A(\mu - 1) + A'(\mu' - 1) \]
where:
- A = angle of the crown prism
- A' = angle of the flint prism
- μ = refractive index of crown glass
- μ' = refractive index of flint glass

### Step 2: Express the refractive indices in terms of dispersive power
Using the dispersive power formula:
\[ \omega = \frac{\mu_B - \mu_R}{\mu - 1} \]
We can rearrange this to find:
\[ \mu - 1 = \frac{\mu_B - \mu_R}{\omega} \]

For crown glass:
\[ \mu - 1 = \frac{0.015}{0.03} = 0.5 \]

For flint glass:
\[ \mu' - 1 = \frac{0.022}{0.05} = 0.44 \]

### Step 3: Substitute into the deviation formula
Now substitute these values back into the deviation formula:
\[ 2 = A(0.5) + A'(0.44) \]

### Step 4: Rearranging the equation
This gives us our first equation:
\[ 2 = 0.5A + 0.44A' \]
Let’s call this Equation (1).

### Step 5: Relate the angles of the prisms
From the condition of no dispersion, we can write:
\[ \mu_B - \mu_R = A'(\mu_B' - \mu_R') \]
Substituting the known values:
\[ 0.015 = A' \times 0.022 \]

### Step 6: Solve for A'
From the above equation:
\[ A' = \frac{0.015}{0.022} \]
Calculating this gives:
\[ A' \approx 0.6818 \]

### Step 7: Substitute A' back into Equation (1)
Now substitute A' back into Equation (1):
\[ 2 = 0.5A + 0.44(0.6818) \]
Calculating the second term:
\[ 0.44 \times 0.6818 \approx 0.29999 \]
So we have:
\[ 2 = 0.5A + 0.3 \]
Rearranging gives:
\[ 0.5A = 2 - 0.3 \]
\[ 0.5A = 1.7 \]
\[ A = \frac{1.7}{0.5} = 3.4 \]

### Step 8: Calculate A'
Now substituting A back into the expression for A':
\[ A' = 0.6818 \]

### Final Values
- Angle of crown prism (A) ≈ 3.4°
- Angle of flint prism (A') ≈ 0.6818°

### Summary
The angles of the two prisms for a deviation of 2° without dispersion are:
- Crown prism angle (A) ≈ 3.4°
- Flint prism angle (A') ≈ 0.6818°