76 BULLETIN OF THE UNIVERSITY OF WISCONSIN 



the quantities m and n be eliminated by means of the rela- 

 tions (78), l we have the following: 



sin c + cos z sin b sin z cos b sin (a + A) = (2) 



where 90 a and b represent the azimuth and altitude of 

 the point in which the rotation axis of the instrument, 

 produced toward the west, intersects the celestial sphere. 

 A and z are the azimuth (reckoned from the north toward 

 east) and zenith distance of a star at the instant of its 

 transit over a thread whose collimation is c, i. e. the point 

 90" a, b is the pole of the small circle traced upon the 

 celestial sphere by the thread in question when the instru- 

 ment is rotated about its axis, and the distance of this 

 circle from its pole equals 9(P + c. 



Since in practice b and c are never so great as 10', equa- 

 tion (2) may be written without sensible loss of accuracy : 

 c + cos z . 6 = (a -J- A) sin z (3) 



Substituting in this equation for b its value as given by 

 the spirit level, and writing a similar equation for the case 

 in which the object observed is not the star, but its image 

 reflected from mercury or some other level surface, we 



have: 



Dir. c' + cos z' (n' T -f- /3) = (a + A') sin z' (4) 



Eef. c" cos z" in" T + /3) = (a +A") sin z' 

 where n' and n" are the measured inclinations of the axis 

 expressed in half divisions of the level scale. We now put 



z' = z + x z" = z x 



and introducing these values into(4) find by substraction : 



c' c" + (n 1 + n') cos x cos Z.T -f- 2 cos x cos z fi 



= (A' A"} cos x sin z + (2a + A' -f- A") sin x cos z (5) 



In practice the object observed will usually be a circum- 

 polar star, and owing to its slow motion the quantity 

 x = J (z' z") will be so small that we may assume 

 cos x = 1 sin x = cos S sin t sin ^ ( T T'} 



where T' and T" are the observed times and t is the hour 

 angle of the star at the instant (T' + T"). 



* Loc. cit. 



