Tacheometry – inclined sights correction: For horizontal distance by stadia with line of sight inclined at angle θ (staff vertical), the standard horizontal-sight formula can be adapted by multiplying which constants?

Introduction / Context:
In stadia (tacheometric) surveying with a vertical staff, the familiar horizontal-sight relation is D = k * s + c, where D is horizontal distance, s is staff intercept, k is the multiplying constant, and c is the additive constant. When sights are inclined by angle θ, appropriate trigonometric factors modify these constants to recover horizontal distance directly.

Given Data / Assumptions:

• Staff is held vertical; line of sight makes angle θ with the horizontal.
• k and c are instrument constants determined by focusing system and stadia interval.
• We seek the formula for horizontal distance (not slope distance).

Concept / Approach:
For inclined sights with vertical staff, the slope-distance relation is S = k * s + c. The horizontal component is D = S * cos θ, but the staff intercept seen by the inclined line of sight also projects with a cos θ factor. Combining these effects yields the well-known result D = k * s * cos^2 θ + c * cos θ, which corresponds to multiplying k by cos^2 θ and c by cos θ in the horizontal-sight formula.

Step-by-Step Solution:

Verification / Alternative check:
Textbook derivations using geometry of similar triangles for the stadia diaphragm with vertical staff confirm the cos^2 θ and cos θ factors for horizontal distance.

Why Other Options Are Wrong:

• sin θ or sin^2 θ factors apply to vertical component computations, not horizontal distance.
• Using a single cos θ for both constants underestimates horizontal distance.
• Applying cos^2 θ to both constants overcorrects the additive part.

Common Pitfalls:
Confusing formulas for staff held normal to the line of sight; forgetting to distinguish slope distance from horizontal distance; mixing up the roles of k and c.

Final Answer:
Multiply the multiplying constant by cos^2 θ and the additive constant by cos θ