d4 



This is Jyaplace's fundamental equation (3) vid. Mec. Cel. 

 torn, 4, \). 244. b here corresponding to -^ iu Laplace's 

 formula. 



2. The integral of this equation from g = (f ) to § = 

 gives the atmospherical refraction required. It is obvious 

 tliat to obtain the complete integral, it is necessary to know 

 the relation between r and §, or the law of diminution of the 

 density of the atmosphere. This is at present unknown ; 

 but notwithstanding, we can approximate sufficiently to the 

 value of R for all values of ^ less than about 80°. 



From the zenith to 74° zenith distance the result is the 

 same whether we approximate to the integral, without know- 

 ing the relation of r and §, or whether we assume any given 

 relation, and reduce equation (2) to a convenient form for 

 fipding the integral. 



Also by assuming two certain laws of variation of density 

 we may obtain two integrals, one of which must give the re- 

 fraction greater than the truth, and the other less. We find 

 that as far as 80° 45', * these refractions do not differ by one 

 second, therefore a mean of the two must always give the 

 refraction true within half a second so far from the zenith. 



* The apparent zenith distance of the bright star, Capella, when below the pole, is 

 in this latitude =i80''45', and having made many observations of this star S. P, I have 

 taken that zenith distance as a limit. 



