**Solution:**

To find the value of R1, we can use the principle of Wheatstone bridge. In a balanced Wheatstone bridge, the ratio of resistances in one arm is equal to the ratio of resistances in the opposite arm.

**1. Initial Condition:**

In the initial condition, the unknown resistance R1 is connected in series with a 10 ohm resistance. Let the resistance connected in the other gap be R2. The balance point is at 50 cm.

Let the length of the meter bridge wire be L. The length from one gap to the balance point is l1 = 50 cm = 0.5 m.

**2. Balance Condition:**

In the balanced condition, the ratio of resistances in the two arms is equal. Therefore, we have:

R1/10 = R2/l2 (1)

where l2 is the length from the other gap to the balance point.

**3. Removal of 10 ohm resistance:**

When the 10 ohm resistance is removed, the balance point shifts to 40 cm.

Let the new length from the first gap to the balance point be l1' = 40 cm = 0.4 m.

**4. New Balance Condition:**

In the new balanced condition, the ratio of resistances in the two arms is still equal. Therefore, we have:

R1/R2 = l1'/l2' (2)

where l2' is the new length from the other gap to the balance point.

**5. Ratio of Lengths:**

From equations (1) and (2), we can equate the ratios of resistances and lengths:

R1/10 = l1'/l2' (3)

**6. Finding R1:**

To find the value of R1, we need to find the values of l1' and l2'.

From the given information, we know that l1 = 0.5 m and l1' = 0.4 m.

To find l2', we can use the fact that the sum of the lengths from the two gaps to the balance point is equal to the length of the meter bridge wire:

l1 + l2 = L (4)

l1' + l2' = L (5)

Subtracting equation (5) from equation (4), we get:

l2 - l2' = l1 - l1'

l2' = l2 - (l1 - l1')

Substituting the given values, we have:

l2' = l2 - (0.5 - 0.4) = l2 - 0.1

**7. Substituting Values:**

Substituting the values of l1', l2', and l1 into equation (3), we have:

R1/10 = 0.4/(l2 - 0.1)

R1 = 4/(l2 - 0.1)

**8. Conclusion:**

The value of R1 in ohms is given by R1 = 4/(l2 - 0.1), where l2 is the length from the other gap to the balance point.