Which of the following errors is/are eliminated by taking both left and right face observations in a theodolite?
Show Hint
Face-change (transiting) cancels errors that reverse sign between face left and face right, such as collimation and trunnion-axis maladjustment, but not centering or graduation errors.
Error due to line of collimation not being perpendicular to the horizontal axis
Error due to the horizontal axis not being perpendicular to the vertical axis
Error due to eccentricity of the verniers
Error due to imperfect graduations
Show Solution
The Correct Option isA, B
Solution and Explanation
Step 1: Classify theodolite errors into three broad families. Systematic theodolite errors generally fall into (i) maladjustment errors, where an axis is not exactly perpendicular to another reference axis, (ii) centering/eccentricity errors of the graduated circles and verniers, and (iii) graduation errors from imperfect circle manufacturing. Each family requires a different field technique to cancel it. Step 2: Show algebraically why face-change cancels a maladjustment error. If a maladjustment introduces a fixed angular error $e$ into the face left reading, $R_{FL} = R_{true} + e$, transiting the telescope reverses the geometric sense of that same misalignment, giving $R_{FR} = R_{true} - e$. Averaging the two:\[ \frac{R_{FL}+R_{FR}}{2} = \frac{(R_{true}+e)+(R_{true}-e)}{2} = R_{true} \]so the error $e$ cancels out exactly. Both the collimation error (option A) and the horizontal-axis error (option B) are maladjustment errors of this reversible type, so both are removed by the face-left/face-right mean. Step 3: Show why vernier eccentricity does not follow this pattern. Eccentricity of the verniers is a fixed offset of the graduated circle's centre from the true rotation axis; this offset does not reverse sign when the telescope is transited, it appears the same way regardless of which face is used. It is removed instead by reading both verniers A and B, diametrically opposite each other, and averaging them, so option (C) is unaffected by changing face. Step 4: Show why graduation error does not follow this pattern either. Imperfect graduation is a local irregularity fixed to specific marks on the circle; changing face does not move the pointer to a different part of the circle, so this error is not cancelled by face-change. It is instead minimized by distributing multiple settings around the circle. Step 5: Conclude. Face left/face right observation cancels only the two maladjustment-type errors in options (A) and (B).\[ \boxed{\text{(A) and (B)}} \]
