Hagen.] -^t)© [Feb. 3, 



instrument and the piers on which it rests, hence, the distance of the arti- 

 ficial horizon varying with the zenith distance of the object observed, this 

 part of the inclination a will be a function of the zenith distance, while the 

 rest as well as tlie inclination y5 will be the same for the same azimuth. 

 Now it will not be difficult to convince onesGl? that the inclination a cannot 

 influence but the observation of zenith distances and the inclination /9 but that 

 of azimuths and hour-angles. Nor is it difficult to foresee, that the inclina- 

 nation a will have a similar eflcct as the flexure of the telescope and gradu- 

 ated circle on account of their gravity, while the inclination [i is compara- 

 ble to the inclination of the horizontal rotation axis to the true horizon. 

 The former two are functions of the zenith distance and may therefore be 

 represented by periodic series, whose terms involve the sines and cosines 

 of its multiples, while the latter two are merely functions of the azimuth. 



Part I. — Influence of the inclination /? on Azimuth- and Hour-angle Ob- 

 servations. 



We shall first suppose any altitude and azimuth instrument exactly ad- 

 justed so that the axis of collimation describes a great circle passing through 

 the true zenith, and consider the influence exerted by the inclination of 

 the artificial horizon on observations by reflection. 



1. Fundamental Formulas. 



If C denotes the point, in which the axis of collimation produced towards 

 the eye-piece meets the celestial sphere, and Z the true zenith, the arc ^S 

 will be perpendicular on the vertical plane C Z in the point Z. (Fig. 1.) 



FIG.1, 



% 



^' 



Again if through the end of the arc fi and through C a great circle is put, 

 the observed object S will be in this circle in the moment, when its re- 

 flected image passes over the middle thread of the telescope. From S let 

 a perpendicular be drawn on the vertical plane of the instrument, whicli 

 may be intersected in S\ and let S and Z be joined by the arc of a 

 great circle. Finally, let the small angles at Z and C be denoted respec- 

 tively by d ^ and C, and /9 be taken positively right-hand of the sbserver. 

 Then we are not to forget, that Z C = Z S, i. e., equal to the true zenith 

 distance z of the observed object in the moment of observation. Now in 

 the isosceles triangle S Z C we have 



cos z = cot C sin d A — cos z cos d A, 



