and on Astronomical Refractions. 4-89 



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 



1 + 2 K p' 

 K p a sin 



1 -} Z &. p v 1 2 a co 



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

 Phil. Trans.., 1838f> and which are equivalent to similar equations 

 given ia the Mec. Celeste. 



, sin d co 



a . 6 = , " =. 



/ f % ,,2\ 



(1 2aco) A/ COS 2Q + /_ j. _\ (i _2co) 2aeo 

 \ a tt J 



= a i u. i being a constant and u a certain function of the 



a 



density, which depends upon the constitution of the atmosphere, 

 and which for the present may remain undefined. 



3 z 2 w 9 + &c. 

 a a 



a sin 9 d co { 1 + 2 a co + &c. } 



d . 80 = 



>V/cos 2 3 + 2 j u + 3 i M 2 + &c. 2 a co 

 a . . 



co=l -- v 



2 * M + 3 z 2 * + &c. 2 co = 2 # + 3 a; 2 . 

 " The quantities rejected being plainly of no account relatively to 



those retained. Further, because co is always less than 1, - - 



1 2 co 



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

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

 Trans. J838. p. 205.) 



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

 [t 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.] 



| Laplace introduces the same simplification. Mec. CM., vol. iv. p. 24. 



