LIMIT OF THE ATMOSPHERE KERBER 



47 



say, all rays that in vacuo are directed towards h, diverge after their 

 passage through the atmosphere from this same point h. If, for 

 instance we imagine a planetary nebula between any iixed star 

 and the earth and the star-like image of the nebula located at the 

 point h (see fig. 3), ymich is now to be considered as the luminous 



fig. 3. 



object seen through our atmosphere, then will the rays ab converg- 

 ing toward h be so refracted by the atmosphere, that on their 

 entrance into the last medium 11' they will appear to diverge from 

 h in the direction cd, and hence an identical virtual image should be 

 seen at that same point (h) at which a real star-like image must 

 have existed if there had been no atmosphere. 



11. 



FIRST APPROXIMATION TO THE HEIGHT OF THE ATMOSPHERE 



By reason of the nature of the curve of the beam of light abed, 

 which has its concave side toward the center of the earth, it is evi- 

 dent that the principal point must lie within the atmosphere. 

 For if e, fig. 3, were the principal point, then a pencil of rays that in 

 vacuum may have the direction ef must necessarily on its entrance 

 into the medium n' go on farther in the direction egk, if an identical 

 image of the object is to be formed at e; but this is impossible be- 

 cause ef and gk cannot be tangents to the same curve at / and g. 

 Hence therefore it follows that H > ch, that is to say, according to 

 the note on equation (5), 



H > 96.3 kilometers 



(6) 



But a more accurate determination results at once from the fol- 



