Two peg, AB = 60 m, near A

Data: A and B are 60 m apart. Setup 1 at the mid-point P: A = 1.585 m, B = 1.287 m. Setup 2 near A, 6 m inside AB (so 54 m from B): A = 1.355 m, B = 1.045 m.

Step 1: True difference of level (instrument at the mid-point)

With equal sight lengths the collimation error cancels, so

$$\text{True difference (A} - \text{B)} = 1.585 - 1.287 = 0.298\ \text{m}$$

Step 2: Apparent difference from the second setup

$$\text{Apparent difference (A} - \text{B)} = 1.355 - 1.045 = 0.310\ \text{m}$$

Step 3: Error and collimation precision

The reading error of a staff is proportional to its distance from the instrument. In the second setup the distances are A = 6 m (near) and B = 54 m (far), so the difference of sight lengths is 48 m.

$$\text{apparent} - \text{true} = 0.310 - 0.298 = +0.012\ \text{m}$$

B is the far staff and is subtracted in (A $-$ B), so a larger apparent difference means the far reading on B is too small: the error in the far reading is $e = -0.012$ m.

$$\text{Error per metre of sight} = \frac{0.012}{48} = 0.00025\ \text{m/m}, \qquad \text{collimation precision} = \frac{1}{4000}$$

This is 25 mm per 100 m of sight, which is more than the usual permissible limit of about 10 mm per 100 m (1 in 10,000) for ordinary levelling, so the instrument needs adjustment.

The readings are too small and the error grows with distance (the far staff reads too little), so the line of collimation is inclined downward.

Step 4: Correct readings for a horizontal line of sight (second setup)

Error in the reading on A $= -0.00025 \times 6 = -0.0015$ m; error in the reading on B $= -0.00025 \times 54 = -0.0135$ m.

$$A = 1.355 - (-0.0015) = 1.3565\ \text{m}, \qquad B = 1.045 - (-0.0135) = 1.0585\ \text{m}$$

Check: A $-$ B $= 1.3565 - 1.0585 = 0.298$ m, equal to the true difference.

Procedure to make the line of collimation horizontal

1. Keep the level at the second setup (near one peg) and centre the bubble with the foot screws.
2. Calculate the correct staff reading for a horizontal line of sight on the far staff (as found above).
3. Using the capstan-headed screws at the top and bottom of the diaphragm, move the horizontal cross-hair (loosen one screw and tighten the other) until it reads the correct value on the far staff.
4. Check that the bubble is still central. If it has moved, centre it with the foot screws and repeat the adjustment.
5. Repeat the two peg test; the error should now be within the permissible limit.

Answer: collimation precision = 1 in 4000 (0.012 m in 48 m, i.e. 25 mm per 100 m), which is more than permissible; the line of collimation is inclined downward. Correct readings for the second setup: A = 1.3565 m, B = 1.0585 m.