50 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 51 



has become less than /„. But whether there are still higher re- 

 flecting layers of air and a still further diminution of the intensity 

 of the twilight is beyond our power of direct observation ; however, 

 the possibility cannot be gainsaid so long as J > 0. 



The fact that the limit of the atmosphere really is considerably 



higher than 79 kilometers, assuming that - = 57.3" is correct, is 



c 



shown by the well-known differential equation for astronomical 

 refractions, 



R , . 



n . sin £ . n' 



R +h 11 



d p - - . - ... (9) 



J n 2 - ( _? _ V n n sin 2 r 

 ^ V R + h ) 



which for small zenith distances reduces to the expression 



« R , 11 



d p > n' r 



R +h u- 



If for h we substitute its maximum value H so that the right-hand 

 side of this expression becomes still smaller, then 



s ^ R , d n 



p > . ri r — 



R + H ^ « 2 



and after integration between the limits n = 1 for the highest layer 

 and n = n' at the earth's surface we have 



whence 



in which 



p 57.3" 

 whence 



H > 



P 



C ' 



R 



n' - 1 = 0.0002820, R = 6366 .7 kilometers 



H > 06.3 kilometers (10) 



which may be compared with the value in (6) above given. 



