Curvature correction in precise levelling — if the horizontal distance L is in kilometres, what is the curvature correction (in millimetres) to apply to a staff reading?

Introduction / Context:
Over long sights in levelling, the Earth’s curvature makes the line of sight diverge from the level surface. The staff reading at distance L therefore includes a curvature effect that must be corrected for precise work. A standard approximate formula expresses this correction in convenient units for field computations.

Given Data / Assumptions:

• L is the sight length in kilometres.
• Only curvature is considered here (refraction is excluded).
• Small-angle approximations are acceptable for standard levelling distances.

Concept / Approach:
The curvature correction, C_c (meters), is approximately 0.0785 * L^2 when L is in kilometres. Expressed in millimetres, this is about 78.5 * L^2 mm; many texts round to 78.4 * L^2 mm. The correction is subtractive from the observed staff reading because the line of sight is above the level surface at the staff position.

Step-by-Step Solution:

Verification / Alternative check:
Typical combined correction (curvature + refraction) is about 67.3 * L^2 mm; the difference confirms that 78.4 * L^2 mm corresponds to curvature alone.

Why Other Options Are Wrong:

• 58.2/64.8/74.8 * L^2 mm: Do not match the standard curvature coefficient; some are closer to combined or unrelated rounded values.

Common Pitfalls:
Mixing curvature-only with combined curvature-and-refraction coefficients; always confirm which correction is being asked.

Final Answer:
78.4 * L^2 mm