217 



[Hagen, 



hence we have for observatious by reflection with the transit instrument 

 in the prime vertical the correction of the hour-angle. 



3/9 



dt = , -. — (14) 



tan z sm ^ ^ ^ 



The meaning of this correction is, that in the moment, when the re- 

 flected image of any object passes the middle thread of this instrument, 

 the actual Jiour-angle of the object observed is 90^ -\- dt or 270'^ -\- dt, /? 

 being positive right-hand of the observer. Yet as also for this instrument 

 the inclination y? of the apparent horizon remains always on the same side, 

 it will be found more convenient to take ^5 positive towards north and 

 consequently to write the correction of the hour-angle as follows : 



3 /? ( + Star west 



~ tan z sin ^ ( — " east. 

 If we now compare this correction with the one for the rotation axis not 

 lying parallel to the horizon, we find them coinciding except their con- 

 stants. Let d denote the sidereal time, when the star passed over the true 

 prime vertical, and T the clock time, when it passed the middle thread of 

 the instrument, and finally, J T the correction of the clock on sidereal 

 time, then the theory of this instrument gives us these formulas for direct 

 observations 



^ = T + JT + t-a^nr^ + siT^ + sinz'siny ^'^' ^^«* 



^ b k c 



rt __ nn _r_ /J 1' _|_ __ " east 



^ "■" tan z sin (p ~ sin cp sin z sin (p 



where b denotes the elevation of the north end of the rotation axis above 



the true horizon, 180° — k the azimuth of this same end, and 90° -|- c its 



angle with the sight-line of the telescope. For observations by reflection, 



180° — z is to be substituted for z, which changes only the sign of b. But 



besides this, the artificial horizon being inclined to the north, the reflected 



image will be observed after the star passed over the prime vertical in the 



west and before it passed over the same in the east. Hence, if we put 



d = 2 (;? + b) as before, the first fraction of the above equations becomes 



b + 2 /? b _ d 



tan z sin ^ "r ^^n z sin 

 b + 2,9 b-d 



Star west 



' tan z sm ^ tan z sm ^ 



Hence the two formulas for the transit instrument in the prime vertical 

 are to be modified for observations by reflection in the following way : 



^ = T -f J T + tanzliny, + ST^ + sin z'sin ^ ^tarwest 

 _ b— d k c 



"T" tan z sin ^ ' sin (p sin z sin ^ eas . . 



Also in this case we shall see, that the constants cannot be found without 



