We have already found the principle of equalising Backsight and foresight distance that if the level is placed exactly midway between two points and staff reading are taken to determine the difference of level then an error is automatically eliminated. But in the case of a river or valley, it is not possible to set up the level midway between two points on the opposite bank. In such a case, the method of reciprocal levelling is adopted.

In reciprocal levelling, the level is set up on both bank of the river or valley and two sets of staff reading is taken by holding the staff on both banks in this case it is found that error is completely eliminated and true difference of level is equal to the mean of the two apparent difference of level.

Procedure:

1. Suppose A and B are two points on an opposite bank of a river. The level is set up very near A and after proper temporary adjustment staff reading are taken at A and B. suppose reading a1 and b1

2. level is shifted and set up very near B and after proper adjustment, staff reading is taken as A and B. Suppose the reading are a2 and b2

Let,

h= true difference of level between A and B

e= combine error due to curvature, refraction and collimation

(Error is +ve and –ve, here error is assumed +ve)

First case:

Correct staff reading A=a1

Correct staff reading B=b1 – e

True difference of level between A and B

h= a1 – (b1-e) ……………………………………………………..eq.1

Second case:

Correct staff reading B=b2

Correct staff reading a=a2 – e

True difference of level between A and B

h= (a2 - e) – b2 ……………………………………………………..eq.2

from (1) and (2)

2h= a1 – (b1-e) + (a2 - e) – b2

2h= a1 – b1 + e + a2 – e – b2

2h= (a1 – b1) + (a2 – b2)

h= [(a1 – b1) + (a2 – b2)] / 2