and on Astronomical Refractions, 489 



Let S A O N be the trajectory described by light emanating from 

 the star S in its passage through the atmosphere to the earth's 

 surface at O, the apparent zenith distance, or the angle which 

 the tangent to the trajectory makes with the line C O K at O, C H 

 perpendicular to S A K, the direction of the ray before it enters 

 the atmosphere = y, a = C O, then 



d.S0= ^ 



V [a + zf-y* 



■ V i + 2Kp' 



1 +2Kp 



r 



a sin 



V 1 — 2 a co 



I assume these equations, which are proved by Mr. Ivory in the 

 Phil. Trans., 1838t 5 and which are equivalent to similar equations 

 given in the Mec. Celeste, 



d.8 





a sin 5 d cjo 



(1 — 2aw) ^/cos 2 a + (— f ~V) (1 — 2aco) — 2«co 

 being a constant and w a certain function of the 



= a i u. 



density, which depends upon the constitution of the atmosphere, 

 and which for the present may remain undefined. 



+ — = 2 i u + 3 i* u 2 

 a' 



&c. 



d.S0 = 



if x = u oo 



i 



a sin d co { 1 + 2 a co + &c. } 

 V cos 2 + 2 i ' u + 3 /- z* 2 + &c. — 2 a co 

 a 



iu — «co = zjr 



__!_ 



2iu + 3 i 2 k 8 + &c.-2«w=2a' + 3 a? 2 . 

 I The quantities rejected being plainly of no account relatively to 



those retained. Further, because to is always less than 1, 



1—2 a co 



is contained between u and a (I + 2 a), and it may be taken equal 



to a, or to the mean value a (1 + «)J«" Thus we have (See Phil. 



Trans. J 838. p. 205.) 



* This quantity must not be confounded with the a. which accompanies 9. 

 [f Mr. Ivory's paper here referred to has been reprinted in L. & E. Phil. Mag., be- 

 ginning in vol. xv. p. 3, and concluding in vol. xvi. — Edit.] 



f Laplace introduces the same simplification. Mec. Cel., vol. iv. p. 24. 



