Stadia Tacheometry – Multiplying Constant in Terms of Optics In a stadia tacheometer, if i is the stadia hair interval (at the diaphragm), f is the focal length of the objective, and d is the distance between the objective and the instrument’s vertical axis, what is the multiplying constant k in the distance formula D = k * s + C?

Introduction / Context:
Stadia tacheometry relates the staff intercept s to the line-of-sight distance D via D = k * s + C, where k is the multiplying constant and C is the additive constant. Understanding the optical origins of k and C helps diagnose instrument behavior and confirms correct constants for field computations.

Given Data / Assumptions:

• i = stadia hair separation (at the diaphragm).
• f = focal length of the objective.
• d = distance from the objective to the instrument axis (adds to additive constant).

Concept / Approach:

From the geometry of similar triangles in the telescope optics, the ratio of image intercept to object intercept is governed by the focal length and the stadia hair interval. This yields D proportional to s with constant of proportionality k = f / i. The additive constant C accounts for f + d depending on the internal focusing arrangement and is independent of k.

Step-by-Step Solution:

Verification / Alternative check:

Calibration sheets for tacheometers list k ≈ 100 for many instruments, consistent with typical f and i values that give f / i ≈ 100.

Why Other Options Are Wrong:

(b) inverts the ratio; (c) is the additive constant, not k; (d) and (e) are unrelated dimensional ratios.

Common Pitfalls:

Confusing multiplying and additive constants; forgetting that k is dimensionless and independent of staff distance.

Final Answer:

f / i