Levelling by Height of Instrument (H.I.) method: check relations Which statement correctly expresses the standard H.I. method relation between back-sights, fore-sights, and the change in reduced level (R.L.) from first to last station?

Introduction / Context:
The Height of Instrument (H.I.) method computes reduced levels by repeatedly establishing the instrument height above a bench or turning point. A simple and powerful check ties the sum of back-sights and fore-sights to the net change in elevation across the run. This is a staple consistency test in levelling books and site practice.

Given Data / Assumptions:

• Back-sight (B.S.) is taken on a point of known or previously computed R.L.
• Fore-sight (F.S.) is taken on a point whose R.L. will be computed next (or a turning point).
• No gross mistakes; readings are consistently entered in the right columns.

Concept / Approach:

From the H.I. method: H.I. = R.L. + B.S., and R.L.(next) = H.I. − F.S. Summing over the run and cancelling intermediate H.I. terms leads to Σ B.S. − Σ F.S. = R.L.(last) − R.L.(first). This provides a straightforward arithmetic check on field notes and helps catch transposition errors before closing the book.

Step-by-Step Solution:

Verification / Alternative check:

In a closed loop where first and last R.L. are the same datum, the check reduces to Σ B.S. − Σ F.S. = 0, a common balancing criterion.

Why Other Options Are Wrong:

Option B is not a recognized identity and mixes symbols inconsistently. Options C and D therefore fail, and E is not generally true.

Common Pitfalls:

Placing a turning-point F.S. in the B.S. column or vice versa; inconsistent bench levels; arithmetic slips causing the check to fail.

Final Answer:

Σ B.S. − Σ F.S. = R.L.(last) − R.L.(first)