Terrestrial Refraction. 549 



The solution o£ (10) is 



l=}+i(^m)> < u > 



■n being tlie value of n at tlie ground where h = 0. 



For h^S km., we may neglect ]i 2 /2H with an error not 

 exceeding 1 in 1500. 



Then, approximately, 



n -n _ gh 

 nn R 



This may be written with an error of less than 1 in 1600, 



ah 

 "o-«=% (12) 



Now by Dale and Gladstone's Law, 



n = yitcr + 1, (13) 



where pu is constant over wide ranges of temperature and 

 pressure for a given gas, and a is the density. Applying 

 this to air, we have, if cr is the density at tlie ground, 



a = CT *-% ( 14 ) 



Thus, the assumption of a constant radius of curvature 

 for a given ray involves a linear law of density, at least for 

 heights up to 15 km. 



In the textbooks — for example, Wink elmann's Handbuch — 

 the assumption is made that the radius of curvature of all 

 rays has the same constant value. This is obviously absurd, 

 tor (6) would then give a law of variation of density with 

 height that depended upon the particular ray utilized ; so 

 that no definite law of density would exist. 



In deriving (14) the radius of curvature has been assumed 

 to be Il 2 /kg. (Substituting the value &=n Rsin <£ , the 

 suffix " " referring to the lower end of the ray, at the 

 ground, we have 



P = R T (15) 



r gn sm </> 



Hence p varies with the ray, being infinite for a vertical 

 ray. 



Phil. Mag. S. 6. Vol. 38. No. 227. Nov. 1919. 2 Q 



