Correction per chain length of 100 links along a slope of α radians is

🔊 Understanding Slope Correction with Angles

This correction is applied when a measurement is made along a slope, and the angle of the slope (\(\alpha\)) is known. The objective is to calculate the true horizontal distance from the measured slope distance. Since the slope distance is always longer than the horizontal distance, this correction is always negative (subtractive).

🔬 Standard Derivation of the Formula

The relationship between the Horizontal distance (H), Slope distance (L), and slope angle (\(\alpha\)) is given by basic trigonometry:

Step 1: Basic Trigonometric Relationship

$$H = L \cos(\alpha)$$

Step 2: Find the Correction

The correction (C) is the difference between the slope distance and the horizontal distance:

$$C = L - H = L - L \cos(\alpha) = L(1 - \cos(\alpha))$$

Step 3: Apply Small Angle Approximation

For small angles typical in surveying, we use the Taylor series expansion for \(\cos(\alpha)\) and take the first two terms:

$$\cos(\alpha) \approx 1 - \frac{\alpha^2}{2}$$

Step 4: Substitute into the Correction Formula

Substituting the approximation back into the correction formula gives:

$$C \approx L\left(1 - \left(1 - \frac{\alpha^2}{2}\right)\right) = L\left(\frac{\alpha^2}{2}\right)$$

Step 5: Apply to a 100-Link Chain

For a standard chain where L = 100 links:

💡 Analysis of the Provided Answer

The standard, mathematically correct formula for the correction is \(50\alpha^2\). However, the answer provided in the source material is \(100\alpha\). This suggests that the question may be based on a non-standard or simplified approximation not typically used in practice.

For your exams, it is critical to know that \(50\alpha^2\) (or more generally \(L\alpha^2/2\)) is the correct derivation. If you encounter this specific question, be aware that the intended answer might be based on a flawed premise, but your understanding of the derivation is what's most important.