Collimation error from 5 m and 45 m setup

Data: A and B are 50 m apart. Mid-point setup F gives the difference of staff readings $B - A = 0.250$ m (B reads more, so B is lower than A by 0.250 m). Setup 2: 5 m from A and 45 m from B; readings A = 1.500 m, B = 1.752 m.

Step 1: Apparent difference and error

$$\text{Apparent difference} = 1.752 - 1.500 = 0.252\ \text{m}$$ $$e = 0.252 - 0.250 = 0.002\ \text{m for the difference of sight lengths } (45 - 5) = 40\ \text{m}$$

Step 2: Magnitude and direction of the collimation error

$$\text{Error per metre} = \frac{0.002}{40} = 0.00005\ \text{m/m} = \frac{1}{20\,000}$$

(that is 5 mm per 100 m, less than the usual limit of 10 mm per 100 m, so the instrument is within tolerance). The far staff B reads too high, so the line of collimation is inclined upward by about 10″ ($0.00005 \times 206265''$).

Step 3: Correct readings with a horizontal line of sight (setup 2)

Error at A (5 m): $0.00005 \times 5 = 0.00025$ m. Error at B (45 m): $0.00005 \times 45 = 0.00225$ m.

$$A = 1.500 - 0.00025 = 1.49975\ \text{m}, \qquad B = 1.752 - 0.00225 = 1.74975\ \text{m}$$

Check: $1.74975 - 1.49975 = 0.250$ m.

Step 4: Collimation precision ratio

$$\text{Precision ratio} = \frac{e}{\text{difference of distances}} = \frac{0.002}{40} = \frac{1}{20\,000}$$

Answer: error = 0.00005 m per metre (1 in 20,000), line of collimation inclined upward; correct readings A = 1.49975 m (≈ 1.500), B = 1.74975 m (≈ 1.750).