terrestrial Refraction. 555 



4. Simpler and more approximate Formula?. 



If, in integrating (19), we assume a linear law for the 

 index ra, i. e. it' we retain only terms of the first degree in h, 

 which is accomplished by putting q — = c, the formula 

 giving A^i becomes : 



A<fc= y [V^MT- V^)j . • . (26) 



h being taken zero, as before. This equation was found to 

 give the same values as (25) for /i^<l km. ; but the values 

 for hi— 2 km. were about 3 per cent, greater than the corre- 

 sponding values of (25), while the discrepancy for /^ = 3km. 

 was about 6 per cent. Formula (26) may therefore be used 

 for heights up to 1 km., to give A</>; t0 the nearest second. 

 The ordinary, simple, Surveyor's Formula, 



2 ~ 2Li> ^ i} 



in which s is the projection of the ray on the earth's surface, 

 is a particular case of (26). For, writing (26) in the form 



** = 7 9^1 / ■ • • • • (28) 



\/a-\-2bhi+ v a 

 we have when 2bh may be neglected in comparison with a, 



A^=^ = ^tan<£ (29) 



\l a 



Replacing A;tan<£ by its approximate value s, we have 



A</>, _ ps 



2 2 ' 



Thus the value we get for the Surveyor's Constant m is 

 given by 



m = pli = -169. 



Formula (27) is derived by assuming the rays of light to 

 be arcs of a circle of constant radius R/m ; and the values of 

 m calculated from observations vary according to the time 

 and locality, ranging from "105 to '168 according to 

 Winkelmann's Handbuclu 



