Bowditch balancing of link traverse XABCY

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

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

Step 2: Closing error

Required change in coordinates from X to Y:

$$\begin{aligned} \Delta N_{true}&=443.610-562.510=-118.900\ \text{m}\\ \Delta E_{true}&=443.610-175.250=268.360\ \text{m} \end{aligned}$$

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

$$\begin{aligned} e_N&=-250.530-(-118.900)=-131.630\ \text{m}\\ e_E&=288.723-(268.360)=20.363\ \text{m}\\ e&=\sqrt{e_N^2+e_E^2}=133.196\ \text{m} \end{aligned}$$

Perimeter (total length) $\Sigma L=461.50$ m, so the relative error is $e/\Sigma L = 1:3.5$.

The permissible closing error is 1:500, i.e. at most 0.923 m, but the error here is 133.196 m (about 1:3.5). The given coordinates of Y and the field data do not agree, so in a real survey this traverse would be rejected and re-measured. The Bowditch adjustment is shown below only to complete the procedure.

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 (443.610 N, 443.610 E).

Answer: adjusted coordinates A (511.884 N, 254.808 E), B (488.781 N, 346.933 E), C (394.648 N, 352.630 E); closing error 133.196 m (1:3.5).