RL, distance, gradient P to Q, instrument RL 1325.750

Reading of the data: the staff is held at right angles (normal) to the line of sight; the anallatic lens gives $C=0$ with $K=100$. No height of instrument is given, so 1325.750 m is taken as the RL of the instrument axis.

Staff normal to the line of sight, angle of elevation $\theta$ (anallatic lens, $C=0$):

$$D=Ks\cos\theta+h\sin\theta,\qquad V=Ks\sin\theta,\qquad \text{RL}=\text{RL of axis}+V-h\cos\theta$$

where $s$ is the staff intercept, $h$ the axial (middle) hair reading, $\theta$ the vertical angle and $K=100$.

Step 1: Horizontal distance and vertical component of each sight

Point P (bearing 84°36', angle of elevation $3^\circ 30'$, $h=2.100$ m):

$$\begin{aligned} s&=2.850-1.350=1.500\ \text{m}\\ D&=(100\times1.500)\cos 3^\circ 30'+2.100\sin 3^\circ 30'=149.720+0.128=149.848\ \text{m}\\ V&=(100\times1.500)\sin 3^\circ 30'=9.157\ \text{m} \end{aligned}$$

Point Q (bearing 142°24', angle of elevation $2^\circ 45'$, $h=2.875$ m):

$$\begin{aligned} s&=3.795-1.955=1.840\ \text{m}\\ D&=(100\times1.840)\cos 2^\circ 45'+2.875\sin 2^\circ 45'=183.788+0.138=183.926\ \text{m}\\ V&=(100\times1.840)\sin 2^\circ 45'=8.828\ \text{m} \end{aligned}$$

Step 2: Reduced levels

RL of instrument axis = 1325.750 m.

$$\begin{aligned} \text{RL}_{P}&=1325.750+9.157-2.100\cos3^\circ30'=1325.750+9.157-2.096=1332.811\ \text{m}\\ \text{RL}_{Q}&=1325.750+8.828-2.875\cos2^\circ45'=1325.750+8.828-2.872=1331.706\ \text{m} \end{aligned}$$

Step 3: Horizontal distance PQ

The angle at A between the two lines is the difference of the bearings, $57^\circ 48'$ (84°36' and 142°24').

$$\begin{aligned} PQ&=\sqrt{D_1^2+D_2^2-2D_1D_2\cos\gamma}\\ &=\sqrt{149.848^2+183.926^2-2\times149.848\times183.926\cos 57^\circ 48'}=164.043\ \text{m} \end{aligned}$$

Step 4: Gradient

$$\text{Gradient}=\frac{\Delta H}{PQ}=\frac{1.105}{164.043}=0.00674$$

The line from P to Q is falling: 1 in 148.5, i.e. 0.67 %, an angle of $0^\circ 23'09''$ with the horizontal.

Answer: PQ = 164.04 m; RL of P = 1332.811 m, RL of Q = 1331.706 m; difference in level = -1.105 m; gradient falling 1 in 148.5 (0.67 %, 0°23').

If the staff is taken as vertical instead, the results are RL of P = 1332.790 m, RL of Q = 1331.693 m, PQ = 163.69 m and a gradient falling 1 in 149.2, very close to the above because the vertical angles are small.