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Tangential tacheometry, both angles depression

Short answer

For both depression angles α1 to lower target and α2 to upper target, H=s/(tanα1−tanα2), V=H tanα2, and RL of staff station = RL of axis − H tanα1 − h, where h is lower target height above the station.

Develop expression for H, V, and R.L. for the tangential system of tachometry when the both sightings are angles of depression.

In the tangential system the vertical angles to two targets (vanes) on a vertical staff, a known distance ss apart, are measured. The staff is below the line of collimation, so both angles are angles of depression.

 O ------------------------------ horizontal
  \ \        alpha2 < alpha1
   \  \_______   upper target B   |
    \  alpha1\__ lower target A   | s
     \                           _|
      <--------- H ------------>  staff station C

Let α1\alpha_1 be the angle of depression to the lower target A and α2\alpha_2 to the upper target B (α1>α2\alpha_1>\alpha_2). Let HH be the horizontal distance, VV the vertical depth of the upper target B below the horizontal line of sight through the instrument axis, and hh the height of the lower target above the staff station C.

Horizontal distance H

Depth of A=V+s=Htan⁡α1Depth of B=V=Htan⁡α2s=H(tan⁡α1−tan⁡α2)H=stan⁡α1−tan⁡α2\begin{aligned} \text{Depth of A}&=V+s=H\tan\alpha_1\\ \text{Depth of B}&=V=H\tan\alpha_2\\ s&=H(\tan\alpha_1-\tan\alpha_2)\\ H&=\frac{s}{\tan\alpha_1-\tan\alpha_2} \end{aligned}

Vertical distance V

V=Htan⁡α2=stan⁡α2tan⁡α1−tan⁡α2V=H\tan\alpha_2=\frac{s\tan\alpha_2}{\tan\alpha_1-\tan\alpha_2}

Reduced level of the staff station

The lower target A is at depth V+s=Htan⁡α1V+s=H\tan\alpha_1 below the axis, and the staff station C is hh below A:

R.L. of C=R.L. of axis−Htan⁡α1−h\text{R.L. of C}=\text{R.L. of axis}-H\tan\alpha_1-h

or, using the upper target, R.L. of C=R.L. of axis−Htan⁡α2−(s+h)\text{R.L. of C}=\text{R.L. of axis}-H\tan\alpha_2-(s+h), where R.L. of axis == R.L. of the instrument station ++ height of instrument.

✓ Answer checked by Claude Opus 5.5 (high effort)

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