Line of sight error over distance: The combined effect of Earth’s curvature and atmospheric refraction on a levelling sight of length L kilometres produces an apparent error (on staff reading or line of collimation) approximately equal to:

Introduction / Context:
In precise levelling, the line of sight is not perfectly horizontal relative to the curved Earth, and atmospheric refraction bends light downward. The net effect causes a systematic apparent error that increases with the square of the sight length. Having a ready constant helps in checks and in two-peg test computations.

Given Data / Assumptions:

• Sight length L is in kilometres.
• Error magnitude is requested in millimetres.
• Standard average refraction coefficient (~0.07) is assumed for typical conditions.

Concept / Approach:

Curvature alone would produce approximately 78.5 L^2 mm rise of the ground relative to a straight line of sight. Refraction (downward bending) partially compensates, often taken as about 1/7 of the curvature effect upward. The commonly used combined correction is about 67.3 L^2 mm (rounded as 67.2 L^2 mm) applied with appropriate sign in computations to remove the bias.

Step-by-Step Solution:

Verification / Alternative check:

Most textbooks and field tables list the combined effect between 67 and 68 L^2 mm depending on assumed refraction; 67.2 L^2 mm is the standard teaching constant for routine calculations.

Why Other Options Are Wrong:

76.3 L^2 mm and 64.5 L^2 mm do not reflect the accepted mean combined value; 6.72 L mm is linear in L and dimensionally inconsistent with the well-known quadratic law.

Common Pitfalls:

Confusing sign conventions; forgetting that equal sight lengths cancel the effect; using metres instead of kilometres for L, causing large numerical errors.

Final Answer:

67.2 L^2 mm