76 BULLETIN OF THE UNIVERSITY OP WISCONSIN 



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

 tions (78),' we have the following: 



sin c-{-cos z sin b — sin z cos b sin (a + .4) =0 (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 colli mation is c, i. e. the point 



90^ a, l) 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 90'' + c. 



Since in practice & and c are nearer so great as 10', equa- 

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



c + cos z.b = (rt 4- --i) siii z (3) 



Substituting in this equation for h 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 in' r A- (i) — (a + .4 ) sin z' (4) 



Ref. c" - cos z" (n" r + fi) = (a + A') siu 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 subtraction: 



c — c" + {n + n") cos x cos z.v -\- 2 cos x cos z (i 



= {A' — A") cos X sin ^ + (2a -f- A' + 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 = ^(z' — z") will be so small that we may assume 



cos X = 1 sill X =- cos d 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 i (2" + T"). 



1 Loc. cit. 



