Two peg: closing error and precision of level

Data: C and D are 50 m apart. Instrument at E (mid-point): C = 1.455 m, D = 1.860 m. Instrument at F with CF = 5 m and DF = 45 m: C = 1.500 m, D = 1.925 m.

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

With equal sight lengths the collimation error cancels, so

$$\text{True difference (D} - \text{C)} = 1.860 - 1.455 = 0.405\ \text{m}\ (\text{D is lower than C})$$

Step 2: Apparent difference from the second setup

$$\text{Apparent difference (D} - \text{C)} = 1.925 - 1.500 = 0.425\ \text{m}$$

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 C = 5 m (near) and D = 45 m (far), so the difference of sight lengths is 40 m.

$$e = \text{apparent} - \text{true} = 0.425 - 0.405 = +0.020\ \text{m}$$ $$\text{Error per metre of sight} = \frac{0.020}{40} = 0.0005\ \text{m/m}, \qquad \text{collimation precision} = \frac{1}{2000}$$

This is 50 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 error is positive: the far staff D reads too much, and the error grows with distance, so the line of collimation is inclined upward.

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

Error in the reading on C $= 0.0005 \times 5 = +0.0025$ m; error in the reading on D $= 0.0005 \times 45 = +0.0225$ m.

$$C = 1.500 - 0.0025 = 1.4975\ \text{m}, \qquad D = 1.925 - 0.0225 = 1.9025\ \text{m}$$

Check: D $-$ C $= 1.9025 - 1.4975 = 0.405$ m, equal to the true difference.

Answer: the closing (collimation) error is 0.020 m in 40 m, that is 0.0005 m per metre or 1 in 2000 (25 mm over a 50 m sight); the line of collimation is inclined upward. The precision of the instrument is 1 in 2000, which is poor, so it needs adjustment. With respect to a horizontal line of sight the readings are C = 1.4975 m and D = 1.9025 m (the readings 1.500 and 1.925 less the errors 0.0025 and 0.0225).