Adjust traverse A-X to E-Y bearings

Note on the data: the table has four angles (A, B, C, D) and three lengths (AB, BC, CD). The last line of the traverse (up to the closing station and its reference line E-Y) has no length or angle, and no coordinates are given for A or for the end station. Hence the final misclosure cannot be numerically completed from the printed data. The method is shown below and every quantity that can be computed from the data is worked out.

Step 1: Bearings from the angles to the right

Angles are measured clockwise from the back station to the forward station, so

$$\text{Bearing of forward line}=\text{Bearing of back line}+\text{angle to the right}\ (\pm360^\circ)$$

With the back line A-X (139°05'45") at A:

• AB = 139°05'45" + 283°50'10" = 62°55'55"
• BC = 62°55'55" + 180° + 256°17'18" − 360° = 139°13'13"
• CD = 139°13'13" + 180° + 98°12'41" − 360° = 57°25'54"
• Line ahead of D = 57°25'54" + 180° + 103°30'34" − 360° = 340°56'28"

Step 2: Angular misclosure and its adjustment

The bearing of the final line (E-Y = 86°20'47") is carried through all $n$ angles of the traverse. The angular misclosure is

$$\alpha=\text{computed bearing of E-Y}-86^\circ20'47''$$

and each of the $n$ angles is corrected by $-\alpha/n$ (the $k$-th bearing by $-k\alpha/n$) before latitudes and departures are computed. The angle at B is marked doubtful, so it should be checked first if $\alpha$ comes out large.

Step 3: Latitudes and departures (bearings before angular adjustment)

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

Step 4: Misclosure in latitude and departure

After including all lines up to the closing station $Z$ with known coordinates $(N_Z, E_Z)$ and $(N_A,E_A)$ for A:

$$e_N=(N_A+\Sigma\text{Lat})-N_Z,\qquad e_E=(E_A+\Sigma\text{Dep})-E_Z,\qquad e=\sqrt{e_N^2+e_E^2}$$

Precision $=e/\Sigma L=1/N$.

Step 5: Adjustment (Bowditch)

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

The corrections are added to each latitude and departure, and the adjusted values are summed from A to get the adjusted coordinates of the stations B, C, D, ... The last computed station then agrees with the known coordinates.