Link traverse B to M, coordinates X and Y

Reference bearings from the coordinates of the control stations:

$$\theta_{AB}=\tan^{-1}\frac{30.000-18.820}{20.000-42.360}=153^\circ 26'06'',\qquad \theta_{MN}=\tan^{-1}\frac{65.667-100.000}{70.600-50.000}=300^\circ 57'50''$$

(The signs of the differences place AB in the south-east and MN in the north-west quadrant.) At B the reference line is BA, with bearing 333°26'06".

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: A (42.360 N, 18.820 E), B (20.000 N, 30.000 E), M (50.000 N, 100.000 E), N (70.600 N, 65.667 E).

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 BA and using the observed angles, the bearing of the last line comes out as 300°59'06", while its known value is 300°57'50" (M-N from the coordinates).

Angular misclosure = computed − known = 300°59'05.8" − 300°57'50.4" = +75.4" ≈ +0°01'15".

The error is spread equally over the 4 angles: correction per angle = −0°00'19". 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 B (20.000 N, 30.000 E), the unadjusted coordinates of M are (49.966 N, 100.025 E), whereas the known coordinates are (50.000 N, 100.000 E).

Step 3: Linear misclosure

$$\begin{aligned} e_N&=49.966-50.000=-0.034\ \text{m}\\ e_E&=100.025-100.000=+0.025\ \text{m}\\ e&=\sqrt{e_N^2+e_E^2}=0.043\ \text{m},\qquad \frac{e}{\Sigma L}=\frac{0.043}{90.025}=\frac{1}{2117} \end{aligned}$$

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 M now equal its known coordinates, so the traverse is balanced.

Answer: X (10.014 N, 59.980 E); Y (29.932 N, 69.929 E) (linear misclosure 0.043 m, 1 in 2117).