104 BULLETIN OF THE UNIVERSITY OF WISCONSIN 



The first of these equations will usually be the more con- 

 venient. 



To determine P we have from the triangle formed by the 

 two stars and the pole 



cos p = cos it sin <$ -}- sin it cos S cos ( 2 a^) 

 where 8 is the declination of the southern star. In place 

 of this rigorous equation we may write with sufficient pre- 

 cision 



p = 90 - 8 - Tt cos (o- 2 - a x ) = H (t) - d (22) 



where the symbol E (t) denotes the tabular value of H cor- 

 responding to the argument t = # 2 <*i 



Equations (18), (20) and (22), in connection with the tab- 

 ular values of E and Jf , suffice for the construction of an 

 observing list, but if any considerable number of stars are 

 to be observed in the same latitude it will be found an 

 economy of labor to construct for the given latitude special 

 tables, such as those given below for the Washfturn Ob- 

 servatory, which are based on the following analysis : 



Neglecting terms of the order n- we put 



cos N = tan cp tan | p 

 and find from equations (18) 



cos N = cos NQ + a cos t cos N 

 N = N Q Tt cos t sec S cot N 

 M = it sin t sec d 



The factor * sin t = J/ has been tabulated, and * cos t is 

 evidently equal to the tabular value of M which corres- 

 ponds to the argument t 6 7i . Putting 



sec d = h sec 8 cot N = k 



we tabulate N , h and k with the argument s and find for 

 the instant at which the two stars have equal altitudes 



T = a + N + h J/'o + k M\ 



the accents ' " denoting that the arguments for the cor- 

 responding values of M are t and t Q h . 



It should be noted that since N is an approximation to 

 the N of the rigorous formulae we shall have N and k posi- 



