Link traverse Q to X, Gale's table m1, m2

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: Q (63148.201 N, 7492.189 E), X (63321.618 N, 7439.604 E); bearing PQ = 61°06'56", bearing XY = 288°07'21".

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 QP (back bearing of PQ = 241°06'56") and using the observed angles, the bearing of the last line comes out as 288°07'36", while its known value is 288°07'21" (X-Y).

Angular misclosure = computed − known = 288°07'36" − 288°07'21" = +0°00'15".

The error is spread equally over the 4 angles: correction per angle = -0°00'04". 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 Q (63148.201 N, 7492.189 E), the unadjusted coordinates of X are (63321.605 N, 7439.540 E), whereas the known coordinates are (63321.618 N, 7439.604 E).

Step 3: Linear misclosure

$$\begin{aligned} e_N&=63321.605-63321.618=-0.013\ \text{m}\\ e_E&=7439.540-7439.604=-0.064\ \text{m}\\ e&=\sqrt{e_N^2+e_E^2}=0.065\ \text{m},\qquad \frac{e}{\Sigma L}=\frac{0.065}{208.462}=\frac{1}{3211} \end{aligned}$$

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

Answer: m1 (63192.407 N, 7470.863 E); m2 (63239.826 N, 7414.348 E) (linear misclosure 0.065 m, 1 in 3211).