1882.] ^l-*- [Hagen. 



For Zg may be taken the mean value of the two nearly equal zenith dis- 

 tances z and z^ and if the instrument had no vertical circle, it may be 

 computed from the declination, the latitude and the mean hour angle. 

 A^ain we have 



-' — =^,-(^^-^)' 



where 1- denotes the variation of the azimuth in the unit of time for the 



dt 



moment ^ (0^ -\- 0). 



Thus far it has been shown, how to find the value of d for one single 

 azimuth, but it will be necessary to have the means of computing it for 

 any azimuth. From the theory of the azimuth instruments it is known, 

 that b is represented by the formula 



b =^ i — iu cos (A — Jo).- 

 where i denotes the inclination of the horizontal axis to the azimuth cir- 

 cle, i„ the inclination of this circle to the true horizon, while A is the azi- 

 muth of the observed object and A„ a constant explained by the formula 

 itself The inclination ,5 of the artificial horizon may be represented by a 

 similar formula 



o = — i^sm(A—Ai), (6) 



where i, is the constant deviation of the plumb line caused by local irregu- 

 larities in the figure and density of the earth, Ai the azimuth of its direc- 

 tion and A the azimuth of the observed object. Hence we find 

 id = ;- + b = i — i„ cos (A — Ao) — 1*1 sin (A — Ai) 



= i — cos A O'o cos Ao — ii sin Ai) — sin A (io sin J„ + ij cos Ai) 

 or if we put 



i„ cos Af, — ii sin A^ = i.., cos A-z ] c^) 



io sin A^ -\- li cos Ai = ij sin A-2 [ 

 we find by a simple transformation 



i d = y? + b = i — ij cos (A — A;) (8) 



To find the three constants i, i^ and A-^ three observations are sufficient, 

 which may be equally distributed in the usual way. Let dj, d2, dj be the 

 values of d, corresponding to the three azimuths A, A + 130°, A + 240" 

 we find from (8) 



i di = i — i.^ cos (A — A-i) 



i d^ = i + ^ ij cos ( J — A.) + ? 12 sin (A — A^) V^ 

 1 ds = i + T 12 cos (A — Js) — k h sin i^ — J.,) 1/3 

 and by adding and subtracting these equations 

 i =: 1 (di -f d^ + dj) 

 ij cos CJ — J,) = Ht^2 + da — 2di) 



i, sin (.-1 - .J,) = ^ (d,-d,.) (9) 



If therefore either of the methods mentioned before, viz., by the striding 



PROC. AMEK. PHILOS. SOC. XX. 111. 2a. PRINTED APRII. 14, 18S2. 



