.• TT *'"• S, Sin. u 



8in. n ^ 



8in.(Z) +A) 



substituting for sin. A, sin. u. cos. S ; and developing 



sin. n = ?!^ . sin. u —JJ^IA^ 8in.« « + &c. 

 sin. D sin. D. tang.Z)' 



I have given the second terms of these expressions to estimate the 

 accuracy of the first ; but the greatest of them does not exceed 

 0". 3 at 85°. ZD, even in the moon's parallax, and may therefore 

 be neglected, and the equations become 



sin A = sin. u. cos. S (I) 



„ sin. «. sin S. 



sin. n = : — (a\ 



sin. D. (2) 



If we put Brinkley's refraction for u, we evidently obtain R and 

 g, the refractions in PD and AR : it is of the form 



u = »j. tang, z K, tang. z. 



The coefficient « being found by dividing the numbers of Brink- 

 ley's second table by tang, z, as I have done in my copy of them ; 

 therefore 



R=z(nt — x) tang. z. cos. S. 



but putting PE = ^ ; (it is the common auxiliary arc used to find 

 the Z. D from the Polar distance and hour angle) we have 



cos. S. tang. z. =: tang. [D — ^) 



therefore 



R = (m — ^).tatig.{D-0 (») 



or where the trouble of forming the tables of » is not taken 



R. = m. tang. (D — ^) c. cos. S. 



c being the number found in table II. When ^ is known, the cal- 

 culation of the refraction in P. D. is little more difficult than that in 

 altitude. This arc is so useftil that I computed a table of its values 

 for every minute in time of the angle P, up to 6". which was easily 



