and on Astronomical Refractions. 491 



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

 altitude. 



Let 



/cos'g + gi*^ cos2 + 2ix = |V 



V z 



, . « i^a 4 — 2 22 2 cos 9 0+cos 4 



d .27 = a d z x l = T5 . 



4 r 



The integral of d . 8 is to be taken from 



cos , 4/ cos 2 + 2 i x" „ 



z = — == = z' 9 to z = = = z" • 



V i */ i 



Let (t-^) represent the value of (-7- -J at the former of 



(d n a>\ " 

 t — -) its value at the latter, then integrating 



continually by parts, 



/• . . d w j 

 a sin -j — a a- 

 d x 

 \/cos 2 + 2 2 X 



&c. 



The second term is 



3 a sin e 2 .z 2 d a> 



2 (cos 2 + 2ix)* 

 3 01 sin f .„ „ . „ , cos 4 0" 



3 



8f 2 



{.**-««-.. + 2*'}^ 



the integral of which is 



3 a sin \d w fi? z 5 . A cos 4 01 

 3- i -5— < — 2*zcos 2 I 



