Steps of closed traverse computation

Computation of a closed theodolite traverse (closed loop) proceeds as follows.

1. Angular check and adjustment: the sum of interior angles must equal $(2n-4)\times90^\circ$. If the angular error is within the permissible limit, it is spread equally over the angles (or according to the weights of the stations).
2. Bearings: starting from the observed or assumed bearing of one line, find the bearings of the others from the adjusted angles. Check that the bearing of the first line is obtained again at the end.
3. Latitudes and departures: $\text{Lat}=L\cos\theta$, $\text{Dep}=L\sin\theta$ for each line.
4. Linear misclosure and precision:

$$e_N=\Sigma\text{Lat},\quad e_E=\Sigma\text{Dep},\quad e=\sqrt{e_N^2+e_E^2},\quad \frac{e}{\Sigma L}=\frac1N$$

Accept the traverse if the precision is within the required limit (e.g. 1 in 5000); otherwise repeat the field work. 5. Balancing: distribute $e_N$ and $e_E$ among the lines by Bowditch's rule (or the transit rule) to get corrected latitudes and departures, whose sums are zero. 6. Final lengths and bearings: from the corrected values, $L=\sqrt{\text{Lat}^2+\text{Dep}^2}$ and $\theta=\tan^{-1}(\text{Dep}/\text{Lat})$. 7. Independent coordinates: starting from the known coordinates of one station (or an assumed origin), add the corrected latitudes and departures in sequence, checking that the starting station is reproduced. 8. Tabulate all the above in a Gale's traverse table, then plot by coordinates.