COMSTOCK STUDIES IN ASTRONOMY 103 



four place logarithms the following approximate equiva- 

 lents of those equations : 



a sin t 



t = or 2 a tan M = 



a = sin TT cosec j) cos N = 



1 a cos 

 tan CD tan 4 



\-acost 

 T! = T 2 == a 2 - AT + M + N 



When the sidereal times ^ and T 2 are known the zenith 

 distances and azimuths of the stars may be directly com- 

 puted from the fundamental formula for the transforma- 

 tion of coordinates, but the following method will usually 

 be found more convenient: 



In the spherical triang]e formed by the polar star, the 

 zenith and the pole, we represent the east hour angle of the 

 star by r and find 



cos z = sin cp sin d^ -f- cos (p cos d^ cos T 



= cos (d 1 <p) cos (p cos Si 2 sin 2 $ r 

 and applying to this the development into series of 



cos x = cos y + h 

 find when terms of the order -H* are neglected 



z = H - <p H = 90 - 7t cos T (19) 



Similarly from the development of the azimuth into 

 series we find when the azimuth is reckoned from the 

 north, positive toward east, 



AI = TC sin r sec (p = Jf sec g> (20) 



Values of H and If "with the argument 'r are tabulated 

 below. 



To determine the difference of azimuth of the stars, we 

 represent by p the length of an arc of a great circle join- 

 ing them, and from the isosceles spherical triangle formed 

 by the two stars and the zenith, find 



cos p = cos-z -f- sin?z cos (A 2 AJ 

 which is readily transposed into either 



sin \ (A 2 A ) =3 sin |- p cosec z 

 or 



sin i p (21) 



tan 



*\sin (z - |) sin (z + 



