Anallatic tacheometer from O, RLs of A and B

Anallatic lens: $C=0$, $K=100$. Zenith angles are converted to vertical angles by $\theta=90^\circ-z$: for A, $90^\circ-85^\circ30'=+4^\circ30'$ (elevation); for B, $90^\circ-101^\circ15'=-11^\circ15'$ (depression). The RL of station O is taken as the ground RL, so the instrument axis is HI above it.

Staff held vertical, line of sight inclined at angle $\theta$ (additive constant zero, which is the case for an anallatic lens or $C=0$):

$$D=Ks\cos^2\theta,\qquad V=\tfrac12Ks\sin2\theta$$

where $s$ is the staff intercept, $\theta$ the vertical angle and $K=100$, $C=0$.

Step 1: Horizontal distance and vertical component of each sight

Point A (bearing 30°30', angle of elevation $4^\circ 30'$):

$$\begin{aligned} s&=2.355-1.155=1.200\ \text{m}\\ D&=100\times1.200\times\cos^2 4^\circ 30'=119.261\ \text{m}\\ V&=\tfrac12\times100\times1.200\times\sin 2(4^\circ 30')=9.386\ \text{m} \end{aligned}$$

Point B (bearing 75°30', angle of depression $11^\circ 15'$):

$$\begin{aligned} s&=2.750-1.250=1.500\ \text{m}\\ D&=100\times1.500\times\cos^2 11^\circ 15'=144.291\ \text{m}\\ V&=\tfrac12\times100\times1.500\times\sin 2(11^\circ 15')=-28.701\ \text{m} \end{aligned}$$

Step 2: Reduced levels

RL of instrument axis = RL of station + HI = 450.800 + 1.550 = 452.350 m.

$$\text{RL of staff point}=\text{RL of axis}+V-h$$ $$\begin{aligned} \text{RL}_{A}&=452.350+9.386-1.755=459.981\ \text{m}\\ \text{RL}_{B}&=452.350-28.701-2.000=421.649\ \text{m} \end{aligned}$$

Step 3: Horizontal distance AB

The angle at O between the two lines is the difference of the bearings, $45^\circ 00'$ (30°30' and 75°30').

$$\begin{aligned} AB&=\sqrt{D_1^2+D_2^2-2D_1D_2\cos\gamma}\\ &=\sqrt{119.261^2+144.291^2-2\times119.261\times144.291\cos 45^\circ 00'}=103.474\ \text{m} \end{aligned}$$

Step 4: Gradient

$$\text{Gradient}=\frac{\Delta H}{AB}=\frac{38.332}{103.474}=0.37045$$

The line from A to B is falling: 1 in 2.7, i.e. 37.05 %, an angle of $20^\circ 19'38''$ with the horizontal.

Answer: AB = 103.47 m; RL of A = 459.981 m, RL of B = 421.649 m; difference in level = -38.332 m; gradient falling 1 in 2.7 (37.05 %, 20°20').