Bowditch link traverse XABCY (X 1877.51 N)

Method: compute latitudes and departures from the given WCBs, find the closing error against the known coordinates of Y, check it against the permissible limit, then distribute it by Bowditch's rule and compute the adjusted coordinates.

Step 1: Latitudes and departures

$$\text{Lat}=L\cos\theta,\qquad \text{Dep}=L\sin\theta$$

(The values for each line are listed in the table in Step 3.)

Step 2: Closing error

Required change in coordinates from X to Y:

$$\begin{aligned} \Delta N_{true}&=1626.500-1877.510=-251.010\ \text{m}\\ \Delta E_{true}&=1578.870-1290.200=288.670\ \text{m} \end{aligned}$$

Computed: $\Sigma \text{Lat}=-250.530$ m, $\Sigma \text{Dep}=288.723$ m.

$$\begin{aligned} e_N&=-250.530-(-251.010)=0.480\ \text{m}\\ e_E&=288.723-(288.670)=0.053\ \text{m}\\ e&=\sqrt{e_N^2+e_E^2}=0.483\ \text{m} \end{aligned}$$

Perimeter (total length) $\Sigma L=461.50$ m, so the relative error is $e/\Sigma L = 1:955$.

Assessment: the permissible error is $461.50/500=0.923$ m, and the actual error of 0.483 m (1:955) is smaller, i.e. better than the permissible 1:500. The traverse is therefore acceptable and the error can be distributed.

Step 3: Bowditch correction

$$C_{lat}=-e_N\frac{L}{\Sigma L},\qquad C_{dep}=-e_E\frac{L}{\Sigma L}$$

Step 4: Adjusted coordinates

Start from X and add the adjusted latitude and departure of each line:

Check: the computed coordinates of Y agree with the given Y (1626.500 N, 1578.870 E).

Answer: adjusted coordinates A (1792.532 N, 1375.039 E), B (1737.511 N, 1472.071 E), C (1605.477 N, 1483.595 E); closing error 0.483 m (1:955), within the permissible 1:500.