M.-P. VOL. I. ] CRA WFORD CONST A NT OF RE PR A CTION. 115 



If now, the southern and northern zenith distances were 

 the same, and if, at the times of observing them, the condi- 

 tions of the atmosphere were the same, the two refractions 

 would be the same, /. e., 



r a =r n . 

 In this case we have 



2r=A B (I) 



In practice these ideal conditions are only approximately 

 satisfied. We therefore proceed as follows: 

 From (7) we have 



2r 8 -r s +r n =A-B (8) 



whence 



2r.= (A B) + (r r.) 

 and 



r. = #(A B) + #(r. r,,)) 



also >- (II) 



r B =#(A B) + tf(r,, r.)) 



In case the northern star is at lower culmination we shall 

 have: 



S n =i8o z n v (9) 



S 8 =z s (10) 



S n +S.=i8o_z u z. (ii) 



^180 [ Z ' n + r n +z' 8 -fr 8 ]. (12) 



Hence 



r n -+r 8 =i8o [z' n +z' B ] -[ + .] (13) 



and 



2 r 8 = i8o-[ Z ' n + Z ' 8 ]-[A+ 8J+ [r. rj. ( 14) 

 Calling 



A'=S n +S 8 (15) 



and since 



B=z'.+z' B (5) 



we have 



r 8=9 o-^ [A'+B] + y 2 [r -r D ] ) 



and > (III) 



r=90 ^[A' + B] + ^[r n rj ) 



In order to obtain the refractions from (II) and (III) it 

 is necessary to know the declinations of the stars, their 

 apparent zenith distances (or rather the sums of the zenith 

 distances of the pairs of north and south stars), and the 

 differences between the refractions of the pairs. The 

 stars chosen for this work are all fundamental, and in a 

 first approximation their declinations are to be considered 



