Omitted measurement, non-adjacent affected sides

When two quantities of a closed traverse are missing, they can be found from $\Sigma\text{Lat}=0$ and $\Sigma\text{Dep}=0$. If the lines with missing data are not adjacent, the known lines lying between them are shifted so that the two unknown lines come together. This is allowed because the closure of a traverse depends only on the lines taken, not on their order.

Procedure

1. Compute latitude $L\cos\theta$ and departure $L\sin\theta$ of every line with complete data.
2. Sum them: $\Sigma\text{Lat}$ and $\Sigma\text{Dep}$ of all the known lines.
3. Draw (or imagine) the known lines placed one after the other, and the two unknown lines at the ends so that the figure closes. The closing line then has

$$\text{Lat}_c=-\Sigma\text{Lat},\qquad \text{Dep}_c=-\Sigma\text{Dep},\qquad l=\sqrt{\text{Lat}_c^2+\text{Dep}_c^2},\qquad \tan\phi=\frac{\text{Dep}_c}{\text{Lat}_c}$$

4. The two unknown lines and this closing line form a triangle (the unknown lines are made adjacent, see the sketch). Solve the triangle:
   • Both lengths missing: use the sine rule, the angles being known from the known bearings.
   • Both bearings missing: use the cosine rule, all three sides being known.
   • One length and one bearing missing: use the sine/cosine rule or the quadratic in the unknown length (two possible solutions; select the one that suits the sketch).
5. Find the missing bearings from the triangle angles, then restore the lines to their original positions. Check by computing $\Sigma\text{Lat}=0$ and $\Sigma\text{Dep}=0$.

 Order in field:  AB(?) BC CD DE(?) EA
 Reordered:       AB(?) DE(?) | BC CD EA (known)

            P
           /\
   AB(?)  /  \  DE(?)
         /    \
        /______\
       Q    l    R
 l = resultant of the known lines BC, CD, EA

If more than two quantities are missing, they cannot be found by calculation and the lines must be re-measured in the field.