Angle misclosure, closed link traverse AT1-ET2

Reading of the data. The first bearing is printed as 329°09′21″ [?]. If 329°09′21″ is taken as the bearing of the line from T1 to A, the computed bearing of ET2 comes out as 95°35′28″, a misclosure of about 10°00′40″, which is impossible for angles measured to the second. The value consistent with the other data is 339°09′21″ for the line from T1 to A (its reverse bearing, from A to T1, is 159°09′21″). That value is used below. The angles are to the right, so the bearing of each forward line = bearing of the previous line + 180° + angle at the station.

Rule (angles to the right): $\text{FB of next line} = \text{FB of previous line} + 180^\circ + \text{angle to the right}$ (add or subtract $360^\circ$ to bring the result between $0^\circ$ and $360^\circ$).

Step 1: Bearings computed from the observed angles

Computed FB of E-T2 = 105°35′28″; given (known) FB of E-T2 = 105°36′08″.

Step 2: Angular misclosure

$$\text{Misclosure} = \text{given} - \text{computed} = 0^\circ 00'40''$$

The computed bearing is 40″ too small. This is only 8″ per angle for 5 angles, which is acceptable for a 20″ theodolite (the permissible value $C\sqrt{n}$ with $C = 20''$ is about $45''$). The error is distributed equally among the 5 angles.

Step 3: Adjustment

Total correction = +0°00′40″ (added to the computed bearing of the last line). It is distributed equally over the 5 observed angles: +0°00′40″ / 5 = +0°00′08″ per angle. The correction to a line is +0°00′08″ multiplied by the number of angles used to reach it.

Check: the corrected FB of the last line equals the given value 105°36′08″.

Answer: angle misclosure = 40″ (computed ET2 = 105°35′28″, adjusted ET2 = 105°36′08″). Adjustment = +8″ per angle (bearing corrections +8″, +16″, +24″, +32″, +40″). Adjusted bearings: T1A 339°09′21″, AB 81°26′47″, BC 168°15′08″, CD 77°32′09″, DE 354°08′22″, ET2 105°36′08″.

Note: the adjusted angles are the observed angles plus 8″ each; the correction is added to each angle to the right because the computed bearing is too small.