48 



SMITHSONIAN MISCELLANEOUS COLLECTIONS 



VOL. 51 



lowing consideration. Since for small zenith distances the curved 

 path of a ] encil of light is very nearly an arc of a circle of very large 

 radius of curvature 7 therefore the tangents (to these curves) bh 

 and ch, fig. 3, can be considered as equal and because for small 

 zenith distances the tangents can be exchanged for the distances 

 of the principal points from the limits of the atmosphere, therefore 

 we have approximately 



H=2 ch 

 or from equation (5) 



H =2R 



ri - 1 

 57.3" 



(1 - A) - 1 



= about 192.6 kilometers 



(7) 



Hence the determination of H from the observations of the twi- 

 light arc as given by Alhazen (leading to the value of 79 kilometers) 

 is far too small, and in fact it follows from Fresnel's formula for the 

 intensity of reflected unpolarized light that the argumentation by 

 Alhazen by no means excludes the existence of still higher strata 

 of air. 



According to Fresnel's formula, if the incident light has the inten- 

 sity unity, and e and b indicate the angles of incidence and refrac- 

 tion, then the intensity of the reflected light is 



J" 



1 sin 2 (* - b) 



2 sin 2 (e + b) 



1 + 



cos 2 (e 



cos 2 (e 



-b)\ 



Since in one case the reflection takes place at the thinner layer 



n 



therefore the refractive index is 



n + dn 



sin b =.(1 + dn) sin e 

 cos b = cos e — sin e tg e 



and by substituting these values we obtain 



and we have 



Sn 



2 



dn 



H 



2 cos 2 e — dn 



or approximately, since e is not far from 90 



j„ = I dn 



\2 cos 2 e 



1 + (sin 2 e — cos 2 e) 7 



(8) 



7 Bruhns: Astronomische Strahlen-brechung, p. 66. 



