139 



Art. XI. ASTRONOMICAL AND NAUTICAL 

 COLLECTIONS. 



No. XV. 



i. An Extension of the Inverse Series for the co?nputa- 

 iion of Refraction, together with a direct solution of the 

 problem. 



Considering the acknowledged and increasing importance 

 oi the accurate determination of astronomical refractions, it 

 may not be thought superfluous to attempt to confirm and ex- 

 tend the mode of computation, which has been adopted for the 

 Table of Refractions printed in the Nautical Almanac, and at 

 the same time to compare its results, in the most unfavourable 

 case for its application, with those of the direct method, which, 

 in that case only, are very readily obtained. 



If r be the refraction, z the density, == 1 — jg 

 y the pressure, x the distance from the centre, 



u the perpendicular falling from the centre on the direction of 



the ray, 

 v the distance of this perpendicular from the point of refraction, 

 s the initial value of w, or ?t v ; we shall have {Coll. VI, VIII,) 



dr = — &y—~ mzdx, 



v 



u = i— 1_£ s = (1 + p — pz) s = (1 + px) s, and, p being a 

 l+pz 



very small fraction, 



tf = & - «» rs a» - a* - 2 PX s, ^=t 



az 



dz _ - v <ty __ — & 



dr " ~ ps dr ps 



^f — & . We may then put, in order the better to observe 

 dr mpsz 



the progress of the subsequent operations, 



