Face observation errors; optical theodolite

Errors eliminated by face observations

Face observation means observing a point with the telescope in the face-left position (vertical circle on the left, FL), then again after transiting the telescope and turning the instrument through 180 degrees (face right, FR), and taking the mean.

Mechanism of elimination

1. Collimation error. On face left the line of sight deviates from the correct direction by a small angle $e$ to one side. After changing face, the deviation is on the opposite side, so one reading is too large by $e$ and the other is too small by $e$. The mean is free of the error.

  FL:   X-------- true line ---------> P
          \_ line of sight (+e)
  FR:   X-------- true line ---------> P
          /_ line of sight (-e)
        mean of FL and FR = true direction

2. Trunnion axis error. If one end of the horizontal axis is higher, the telescope sweeps an inclined plane. After changing face, the high end is on the opposite side, so the angular error changes sign and cancels in the mean. (The error is large only for steep sights.)

3. Index error of the vertical circle. If the vertical circle reads $+i$ on face left when the line of sight is horizontal, the same error has the opposite effect on face right. The mean of the two vertical angles is free of $i$.

For horizontal angles, both verniers (A and B, 180 degrees apart) are also read and averaged, which removes the error due to eccentricity of the centre of the circle.

Principle of operation of an optical (micrometer) theodolite

In an optical theodolite the horizontal and vertical circles are made of glass and are finely graduated (for example, at 20 minute intervals). The images of the graduations are carried by a system of prisms and lenses to a reading microscope placed beside the eyepiece of the telescope. The observer reads the angle through this microscope without moving from the observing position.

• The circle is lit by a mirror (or a lamp), and the light passes through the circle, a prism system and a lens to the reading eyepiece.
• In a micrometer theodolite, the graduations of diametrically opposite sides of the circle are brought into the same field of view, so the reading is automatically the mean of the two sides. This removes the error of eccentricity.
• A micrometer screw turns a plane-parallel glass plate in the optical path. Turning it shifts the image of the circle graduations until the index line coincides exactly with a main graduation line. The drum attached to the screw is graduated in minutes and seconds and gives the fraction of the smallest circle division.

   mirror -> circle -> prisms
        -> micrometer plate -> eyepiece
      (plate turned by micrometer screw)

   reading = circle division + micrometer drum reading

Its advantages are: no verniers, quick and accurate reading to 1 second or even 0.1 second, a lighter and sealed instrument, and elimination of the eccentricity error.