Tangential system expressions, angles of elevation

Tangential system

In the tangential method the staff carries two vanes (targets) at a known vertical distance $s$ apart, and no stadia hairs are used. The vertical angles to the two vanes are measured. The method is used when the diaphragm has no stadia hairs.

Both angles of elevation

Let the instrument be at A with axis at height $h_i$ above the station, and the staff be held at B.

• $\alpha_1$ = angle of elevation to the lower vane, $\alpha_2$ = angle of elevation to the upper vane ($\alpha_2 > \alpha_1$),
• $s$ = vertical distance between the vanes, $h$ = height of the lower vane above the staff foot (or the staff reading of the lower vane), $D$ = horizontal distance AB.

                          * upper vane
                     _.-' |  s
                _.-'      * lower vane
        a2 _.-' a1  . '   |  h
   A _.-'-------------------- B
   |<--------- D -------->|

From the two triangles formed with the horizontal through the instrument axis:

$$\begin{aligned} V_1 &= D\tan\alpha_1 \quad (\text{lower vane}) \\ V_2 &= D\tan\alpha_2 \quad (\text{upper vane}) \\ s &= V_2 - V_1 = D(\tan\alpha_2 - \tan\alpha_1) \end{aligned}$$

Horizontal distance

$$\boxed{D = \frac{s}{\tan\alpha_2 - \tan\alpha_1}}$$

Vertical component and RL

$$V_1 = D\tan\alpha_1 = \frac{s\tan\alpha_1}{\tan\alpha_2 - \tan\alpha_1}$$

The lower vane is at height $h$ above the staff foot B. The level of the vane equals the axis level plus $V_1$:

$$RL_B + h = RL_A + h_i + V_1$$ $$\boxed{RL_B = RL_A + h_i + D\tan\alpha_1 - h}$$

If the staff is held on a BM of known RL and the RL of the instrument station is wanted, $RL_A = RL_{BM} + h - h_i - D\tan\alpha_1$.

Here $RL_A$ = RL of the instrument station, $h_i$ = height of the instrument, and $h$ is read from the staff (or the height of the lower vane).