LIMIT OF THE ATMOSPHERE KERBER 



III. SYMMETRIC POINTS OF THE ATMOSPHERE AND THE ZENITH POINT 



OF THE RAYS 



For brevity I speak of the two conjugate points a and a' in fig. 5 

 as symmetrical wh>en the angles of divergence of the rays from the 

 axis are equal to each other so that 



a = a 



(11) 



Let the distances of these points from the center of the earth be 

 D and D'. According to the theory of Gauss the ratio of the dimen- 



sions of the object and its image, or D/D f , multiplied by the ratio of 



the tangents of the respective angles of divergence ^— is equal to the 



tga' 



inverse ratio of the indices of refraction of the first and last media 8 



whence 



D l i a „•> 



or by equation (11) 



D' tga' 



D 

 D' 



u' 



and by equation (4) 



D 



D' 



F 



F' 



or 



F 

 D 



F' 



(12) 



A second equation between D and D' results from the relation devel- 

 oped by Gauss between two pairs of conjugate points 9 for instance, 

 the nodal point m and the symmetrical points a and a' of fig. 5. 

 If we divide the distances F and F' of any pair of conjugate points 



•Mousson: 2d Edition, section 730 (7); 3d Edition, section 808 (19). 

 •Mousson: 2d Edition, section 730 (4); 3d Edition, section 808 (16). 



