Levelling with inverted and upright sights — when one staff reading is taken below the plane of collimation and another above it, the difference of level equals the sum of the two readings; what fixed staff error is introduced in the result?

Introduction / Context:
In precise levelling, staff readings must reference the same physical zero. If the staff's zero graduation does not coincide exactly with the foot (shoe) of the staff—i.e., if there is a fixed offset a—certain combinations of readings can accumulate this error into the computed difference of level (D.L.). This question concerns the case of one reading below and one above the line of collimation, where D.L. is obtained by adding the two readings.

Given Data / Assumptions:

• Let the fixed offset between the staff foot and the zero graduation be a.
• One sight is an inverted (or ceiling) reading; the other is a normal upright reading.
• The computed difference of level equals the sum of the two scale readings.

Concept / Approach:

If each individual staff reading includes the constant zero offset a (because the true zero is a units away from the physical foot), then when adding two readings, the error a is included twice. Consequently, the D.L. is biased by +2a. Recognizing this helps surveyors avoid mixed-sight techniques with miscalibrated staffs or to apply the proper correction when unavoidable.

Step-by-Step Solution:

Verification / Alternative check:

Staff calibration checks (comparing to a standard) quantify a; the correction of −2a removes the bias in the add-case scenario.

Why Other Options Are Wrong:

Single offset a — underestimates the accumulated bias when two readings are added.
Thrice the distance — no basis for triple accumulation in this configuration.
None — incorrect because a fixed bias clearly exists.

Common Pitfalls:

Ignoring fixed staff errors; mixing inverted and upright sights without awareness of how the zero offset propagates into the D.L.

Final Answer:

twice the distance between the zero of graduation and the foot of the staff