and on Astronomical Refractions. 491 



This series may be used if the atmosphere extends only to a finite 

 altitude. 

 Let 



cos 



= 

 V i 



z 4 2*Vcos 9 0+cos 4 



= zn z 



xr = 



4. i 2 

 The integral of d . 8 is to be taken from 



cos . >/ cos 2 + 2 i a?" 



* = -= =a JT| to Z -==- = Z" 



V i v i 



/d*a>\t Sd n io\ 

 Let ( T - j represent the value of ( ^ n ) at the former of 



(d" co\ " 



-T^j its value at the latter, then integrating 



continually by parts, 



/. . d co , 



/a sin -i d A- 

 da; 

 ..... /cos a + C 2 1 x 



rn 



&c. 

 The second term is 



3 a sin z 2 a* d co 



2 (cos 9 + 2z'.r) 

 3 a sin0 cos 4 0"d 



z* 

 the integral of which is 



3 sin fd a) f~i* a 3 cos 4 01 



3 < j -^ -r 2 * 2 cos 2 I 



8 ;f [d^l 3 J 



