562 Mr. A. R. McLeod on 



7. Examination of the ordinary Formula for Range. 

 Let AB (fig. 5) be a ray of light subtending an angle 7 at 

 the centre, 0, of the earth. Let the zenith distances at its 



Fis. 5. 



extremities be O and £, the suffix i being dropped for con- 

 venience. We suppose the lower extremity, A, to be on the 

 surface of the earth (sea-level), while the upper extremity, B, 

 is at height It. Let co be the refraction correction at the 

 point A; then A</> — ^o is lne refraction correction at B, 

 /\6 beino- the refraction of the ray. Finally, let s be the 

 rano-e, measured on the surface of the earth (the projection 

 of the light ray), and let / be the length of the ray AB. 

 Then (cf. Winkelmann's HandbucJi, Band vi. p. 535) 



CO 



h= 2Rsinl^ 



s^ + w -|) 



(39) 



2 sin (</>o + &> — 7)' 

 The assumptions ordinarily made in simplifying (39) are 

 as follows : — 



(i.) We replace 2R sin 7/2 by s = yTi. 

 (ii.) We assume that the ray is an arc of a circle of 

 constant radius, so^ that &> = A</>~ *>o = ™7/ 2 

 where m is a numerical constant, 

 (iii.) 7 2 is negligible in comparison with unity, 

 (iv.) 7 cot c^is negligible in comparison with unity. 

 These assumptions lead to the well-known formula : 



s 2 (l-m) 

 ] L = s cot $0 + — 2R 



(40) 



