The instrument shifts after the 3rd and 6th readings. How are readings 3, 4 and 6 treated?

3rd is FS on the BM, 4th is BS on the BM from the new set-up, 6th is FS on CP2

3rd is BS, 4th is FS, 6th is IS

What does a negative reading imply, and how is RL found?

Point is above the line of sight; RL = HI + numerical value

Point is below the line of sight; RL = HI + value

What is the arithmetic check for this HI-method book?

ΣBS - ΣFS = last RL - first RL = 5.286 m

2073 Shrawan · 6 marks

Level book, benchmark RL 100.00 at 3rd reading

Short answer

Using height of instrument method, RLs of stations 1 to 7 are 97.490, 96.860, 100.000 (BM), 98.160, 98.130, 99.406, 102.776 m.

The consecutive readings taken during a levelling operation are as follows: 0.685, 1.315, -1.825, -0.635, 1.205, 1.235, 2.631, 1.355, -2.015. The instrument was shifted after the third and sixth readings. The third reading was taken to a benchmark of assumed elevation 100.00. Find the reduced levels of other points.

Arranging the readings. The instrument was shifted after the 3rd and 6th readings, so there are three set-ups. The 3rd reading is on the benchmark (assumed RL 100.000 m), which is also the first change point.

Set-up 1: BS 0.685 on station 1; IS 1.315 on station 2; FS $-1.825$ on the BM (point 3).
Set-up 2: BS $-0.635$ on the BM; IS 1.205 on station 4; FS 1.235 on CP2 (point 5).
Set-up 3: BS 2.631 on CP2; IS 1.355 on station 6; FS $-2.015$ on station 7.

A negative reading means the staff was held inverted against an overhead object: the point is above the line of sight, so RL $=$ HI $-$ (reading) $=$ HI $+$ the numerical value. Because the BM (3rd point) has the known RL, the HI of set-up 1 is $100.000 + (-1.825) = 98.175$ m, and the RLs of points 1 and 2 are found from it. The HI of set-up 2 is $100.000 + (-0.635) = 99.365$ m.

Level field book (height of instrument method)

Station	BS	IS	FS	HI (m)	RL (m)
1	0.685			98.175	97.490
2		1.315			96.860
3 (BM, CP1)	-0.635		-1.825	99.365	100.000
4		1.205			98.160
5 (CP2)	2.631		1.235	100.761	98.130
6		1.355			99.406
7			-2.015		102.776

Arithmetic check

$\Sigma BS - \Sigma FS = 2.681 - (-2.605) = 5.286$ ; last RL $-$ first RL $= 102.776 - 97.490 = 5.286$ . Both are equal, so the reduction is correct.

Answer: RLs of the stations 1 to 7 = 97.490, 96.860, 100.000, 98.160, 98.130, 99.406, 102.776 m (the BM, point 3, being 100.000 m).

✓ Answer checked by Claude Opus 5.5 (high effort)

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