DERIVATIONS 83 



Writing (3.56) in terms of A^ units, 



02(mrad) ^ ^0i + — X lO' - 2iNr - N2) (3.57) 



if 01 is in milliradians. 



Generalizing (3.57) for the ^'th and the (A: + l)st layers, 



0,+i(mrad) ^ xJeUmrsid) + ^"^ ^ X lO' - 2{N, - N,+r). (3.58) 



Also from the geometry shown in figure 3.19, a useful relationship for 

 Ti,2 can be obtained. Tangent lines drawn at A and B will be respec- 

 tively perpendicular to ri and r-z, since ri and r2 describe spheres of refrac- 

 tive indices Ui and n2 concentric with 0. Therefore, 



angle AEC ^ angle AOB ^ <}> 



also, in triangle A EC 



angle ACE = 180° - angle CAE - angle A EC 



= 180° - di - <t>. (3.59) 



But from triangle DCB 



angle ACE = angle DCB = 180° - ti,2 - ^2. (3.60) 



Thus 



180° - ri.2 - 02 = 180° - di - <t>, 

 or 



T1.2 - (^ + (01 - 02). (3.61) 



Now since <f> in radians = d/a, where d is distance along the earth's 

 surface : 



T1.2 = - + (01 - 02), (3.62) 



a 



or the bending of a ray between any two layers is given in terms of the 

 distance, d, along the earth's surface from the transmitter (or receiver), 

 the earth's radius, a, and the elevation angles 0i and 02 (in radians) at the 

 beginning and end of the layer. 



