When measuring area in surveying, using a chain or tape with an incorrect length introduces an error. This error needs to be corrected to obtain the true area. Let's understand how this correction is applied.

Chain Error and Its Effect on Measurement

Suppose the nominal length of the chain (the length it is supposed to be) is $L_{nominal}$, and the actual length of the chain is $L_{actual}$. The error in the chain length is $\delta = L_{actual} - L_{nominal}$. The fractional error 'e' per unit length can be defined as:

$\qquad e = \frac{L_{actual} - L_{nominal}}{L_{nominal}} = \frac{\delta}{L_{nominal}}$

Therefore, the actual length of the chain is $L_{actual} = L_{nominal}(1 + e)$.

When this incorrect chain is used to measure a length, say a recorded length of $L'$, the true length $L$ is proportional to the actual length of the chain relative to its nominal length. The relationship between true length and recorded length is:

$\qquad L = L' \times \left(\frac{\text{Actual length of chain}}{\text{Nominal length of chain}}\right)$

$\qquad L = L' \times \left(\frac{L_{actual}}{L_{nominal}}\right)$

Substituting the relation for $L_{actual}$ in terms of 'e':

$\qquad L = L' \times (1 + e)$

Area Correction Due to Incorrect Chain

Area is a two-dimensional measurement. If a rectangular area is measured with recorded dimensions $L'$ and $W'$ using an incorrect chain, the recorded area $A'$ is $A' = L' \times W'$.

The true dimensions corresponding to the recorded dimensions $L'$ and $W'$ would be $L = L'(1+e)$ and $W = W'(1+e)$, based on the correction derived for linear measurement.

The true area $A$ is the product of the true dimensions:

$\qquad A = L \times W$

Substituting the expressions for $L$ and $W$:

$\qquad A = (L'(1+e)) \times (W'(1+e))$

$\qquad A = L'W'(1+e)^2$

Since $A' = L'W'$, we get:

$\qquad A = A'(1+e)^2$

Expanding the term $(1+e)^2$:

$\qquad (1+e)^2 = 1^2 + 2 \times 1 \times e + e^2 = 1 + 2e + e^2$

So, the true area is:

$\qquad A = A'(1 + 2e + e^2)$

In surveying corrections, especially when the error 'e' is small (which is usually the case for a slightly incorrect chain), the term $e^2$ is very small compared to 'e' and $2e$. For example, if $e = 0.01$ (a 1% error), $e^2 = 0.0001$. Therefore, $e^2$ can often be neglected for practical purposes.

Neglecting the $e^2$ term, the formula for the correction to area becomes:

$\qquad A \approx A'(1 + 2e)$

Where:

• $A$ is the true area
• $A'$ is the measured or recorded area
• $e$ is the fractional error in length, $e = \frac{\text{Actual length} - \text{Nominal length}}{\text{Nominal length}}$

This formula relates the true area to the measured area based on the linear error per unit length of the chain.

Analyzing the Options

Let's look at the given options:

1. $A' = (1 + 4e)A$: This doesn't match our derivation. It also swaps A and A' and has a factor of 4.
2. $A = (1 + 2e)A'$: This matches the derived formula, neglecting the $e^2$ term.
3. $A = (1 - 2e)A'$: This formula would be used if the chain was shorter than nominal (negative 'e') and the $2e$ term was treated as a reduction factor, but the standard derivation leads to $(1+e)^2 \approx 1+2e$. If 'e' is defined as positive for a long chain, this formula would be incorrect for a long chain.
4. $A = (2e - 1)A'$: This formula does not represent a valid correction factor based on the linear error 'e'. Correction factors are typically of the form (1 + correction term).

Based on the derivation considering the square of the linear correction factor and neglecting the $e^2$ term, the formula $A = (1 + 2e)A'$ is the correct one used for area correction due to an incorrect chain.

The final answer is $\mathbf{A = (1 + 2e)A'}$.