Levelling Arithmetic – Expressions for Total Change in Level Along a Line In differential levelling, the total change in level between the first and last stations can be expressed equivalently by which of the following?

Introduction:
Levelling computations can be carried out by either the Rise and Fall method or the Height of Instrument (HI) method. Both lead to equivalent expressions for the overall change in elevation from start to end of a levelling run. Recognizing the equivalence is useful for checks and error detection.

Given Data / Assumptions:

• Back sight (BS) and fore sight (FS) readings are taken properly.
• Intermediate sights (IS) are handled according to the chosen method.
• Reduced levels (RLs) are computed consistently from a known benchmark.

Concept / Approach:

In the HI method, net change in level = ΣBS − ΣFS. In the Rise and Fall method, net change in level = ΣRises − ΣFalls. By definition of RLs, net change also equals RL_last − RL_first. Therefore, all three statements are equivalent ways of expressing the same total elevation difference across the line.

Step-by-Step Solution:

Verification / Alternative check:

Well-kept field books include arithmetical checks ensuring ΣBS − ΣFS = ΣRises − ΣFalls = RL_last − RL_first.

Why Other Options Are Wrong:

“None of the above” is false since all listed relations are standard and equivalent.

Common Pitfalls:

Mixing signs on rises/falls; misplacing an IS as BS or FS; inconsistent RL reference causing mismatched totals.

Final Answer:

All of the above