82 TROPOSPHERIC REFRACTION 



3.12. Derivations 



The approximate relation between ^i and 62 is derived here. This rela- 

 tion holds for small increments of height and small ^'s. The relationship 

 was used in making all sample computations in preceding sections. 

 Since for small ^'s 



cos ^1^1- ~ and cos 02 = 1 - - , (3.50) 



and knowing 



r, = ri + ^h (fig. 3.19) 



then substituting in (3.1) yields 



n,{n + A/i)( 1 - f ) = n:n( 1 - f ) , (3.51) 



or 



i2 



'2 . dl 



UiVi + UiAh — riiri -- — n^Ah -- ^ UiTi — UiVi — . (3.52) 



Dividing by Vi 



. UiAh niQ\ Ah dl ^ e\ ,_ __. 



^2 + -^ - -^ - n^ - 2 ^ ni - n. -- . (3.53) 



Since the term — n2 (A/i/rO {61/2) is small with respect to the other 

 terms of (3.53) it may be neglected, and thus: 



n2Ah b\ ^ d\ 



or 



n2 + -^— - n2 77 = ni - ni -^ (3.54) 



T\ Z Zi 



^2 ~ ^1 ^2A/l . , , . 



-n2 7,- = -ni — + (ni - n2). (3.55) 



If one now divides both sides of (3.55) by ni and assumes 

 (ni — ri2)/r?2 = Wi — 712 and nxln-i = 1, (3.55) may be arranged to yield 



72 ^ x/^i + -— - 2(ni - 71,). (3.56) 



