The Sun as a position finder:Theodolite observations

This exercise involves the use of a theodolite to obtain the azimuth of the Sun at a particular time. Knowing the hour angle of the Sun at the time of the observation, the angle between the Sun and the meridian can be calculated. The position of the meridian may, in turn, be related to the observer by means of the azimuth of a fixed reference object and it is then determined once and for all.
Check that the theodolite is level and adjust if necessary. Having taken the precaution of checking that the filters are fitted to the telescope, direct the theodolite towards the Sun and record the following times:
• the moment when the limb of the Sun approaching the cross-wire makes contact with it; and
• the moment when the opposite limb breaks contact with the cross-wire, after the solar image has trailed across the field of view.
The mean of these two observations gives the time at which the centre of the Sun’s disc was at the cross-wire. Note the angle on the horizontal circular scale of the theodolite at which the Sun was observed. Note also the angle of some distant fixed reference object.

Figure 1. Determination of the Sun’s azimuth
Now the situation is depicted in figure 1 showing that the apparent azimuth of a celestial object is not altered by the effect of refraction.
By using the sine formula, we have

giving

Also, we have

giving

Hence, elimination of sin z gives