Reciprocal levelling between two pegs X and Y using two instrument positions: From near X, staff readings on X and Y are 1.755 m and 2.850 m. From near Y, staff readings on Y and X are 0.655 m and 1.560 m. If the reduced level (R.L.) of X is 105.500 m, compute the true R.L. of Y (reciprocal method eliminates collimation and curvature/refraction).

Introduction / Context:
Reciprocal levelling eliminates line-of-sight (collimation) error and the combined effect of curvature and refraction by observing both points from two positions near each point. This is essential when long sights or obstacles prevent equal sight lengths. The problem asks for the true R.L. of Y given four staff readings and the known R.L. of X.

Given Data / Assumptions:

• From near X: r_X@X = 1.755 m, r_Y@X = 2.850 m.
• From near Y: r_Y@Y = 0.655 m, r_X@Y = 1.560 m.
• R.L.(X) = 105.500 m.
• Standard reciprocal-levelling relations apply.

Concept / Approach:

Let A = X and B = Y. The true difference in level (B relative to A) using reciprocal readings is obtained by averaging the two observed differences with appropriate sign. A compact relation is:

This cancels the collimation error term (proportional to sight length), leaving the true elevation difference unaffected by curvature/refraction as well.

Step-by-Step Solution:

Verification / Alternative check:

Compute Y by each setup ignoring collimation, then average: From near X → Y = 105.500 + 1.755 − 2.850 = 104.405 m; From near Y → Y = 105.500 + (1.560 − 0.655) = 106.405 m; Average = 105.405 m, matching the reciprocal formula.

Why Other Options Are Wrong:

105.000 m and 104.500 m underestimate; 105.800 m overestimates; 104.000 m is far from both independent estimates.

Common Pitfalls:

Mixing the sign in the averaging formula; averaging the four readings directly; forgetting that larger staff reading indicates lower ground (for the same H.I.).

Final Answer:

105.405 m