Link traverse Gale's table M2 to M8

A link traverse is checked at both ends, so the adjustment has two stages: (1) balance the angles using the known bearing of the last line, (2) balance the coordinates using the known position of the last station.

Known data: M2 = (1000 N, 1000 E), M8 = (1095 N, 1042 E); bearing M1-M2 = 105°, bearing M8-M9 = 36°.

Step 1: Bearings and angular misclosure

The angles are measured clockwise. Bearing of a line ahead = bearing of the line behind reversed (back bearing) + the clockwise angle at the station. Starting from the line M2-M1 (back bearing 285°00', since M1-M2 = 105°) and using the observed angles, the bearing of the last line comes out as 35°00', while its known value is 36°00' (M8-M9).

Angular misclosure = computed − known = 35°00' − 36°00' = -1°00'.

The error is spread equally over the 4 angles: correction per angle = +0°15'. The k-th bearing therefore receives k times this correction.

Step 2: Latitudes and departures

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

Starting from M2 (1000.000 N, 1000.000 E), the unadjusted coordinates of M8 are (1095.149 N, 1041.981 E), whereas the known coordinates are (1095.000 N, 1042.000 E).

Step 3: Linear misclosure

$$\begin{aligned} e_N&=1095.149-1095.000=+0.149\ \text{m}\\ e_E&=1041.981-1042.000=-0.019\ \text{m}\\ e&=\sqrt{e_N^2+e_E^2}=0.150\ \text{m},\qquad \frac{e}{\Sigma L}=\frac{0.150}{132.850}=\frac{1}{883} \end{aligned}$$

Step 4: Adjustment by Bowditch's rule

Bowditch's rule: the correction to each line is proportional to its length:

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

Step 5: Final coordinates

The computed coordinates of M8 now equal its known coordinates, so the traverse is balanced.

Answer: m1 (1028.951 N, 1020.494 E); m2 (1060.041 N, 1065.829 E) (linear misclosure 0.150 m, 1 in 883).