Hieory of Stellar Scintillation. 141 



If c be neglected altogether, we fall back upon the former 

 equations (19), (20). For the purposes of a second ap- 

 proximation c, though it cannot be neglected, may be calcu- 

 lated as if the refraction were small, and the curvature of the 

 strata negligible. If r) be the whole linear deviation of the 

 ray due to the refraction, 



c=*7/sin0, (22) 



and, as in (16), 



77 = (^ o -l)Zsin0/cos 2 0, .... (23) 



so that 



o= ( ^^ (24) 



cos- 



By equations (21), (24) the value of 86 may be calculated 

 from the trigonometrical tables without further approximation. 

 To obtain an expansion, we have 



86>=sin8(9= ^° tan / 6> -tan<9cosS0 

 1 + c/a 



=0*0-1) tan d{ 1 - (~^y a + Wo ~ 1) tan 2 0~j 



=Gio-i)(i-i)w 



_fo-l)(L_«£=IJW*. • • • (25) 



To this order of approximation the refraction can be expressed 

 in terms of the condition of things at the earth's surface, and 

 (25) is equivalent to an expression deduced at great length 

 by Laplace. 



From the value of I already quoted, and a = 6*3709xl0 8 

 centim., we get 



Z/a=-0012541 (26) 



If further we take as the value under standard conditions for 

 the line D 



^ -1 = -0002927, (27) 



we find as the refraction expressed in seconds of arc 



S0 = 6O''-29tan0-O"-O6688tan 3 <9. . . . (28) 



In (28) is the apparent zenith distance, and it should be 



