58 



itiations which require correction for each individual ob- 

 servatory. 



For about 74° from the zenith, the refraction is inde- 

 pendent of the law of density, and requires only an exact 

 knowledge of the air's refractive power ; this, however, has 

 not been yet obtained with sufficient accuracy by direct 

 experiment, and, therefore, must be deduced from astrono- 

 mical observations. At greater zenith distances some con- 

 stitution of the atmosphere must be assumed, and if its 

 expression contain a sufficient number of arbitrary constants, 

 the resulting refraction can always be made to represent 

 with sufficient exactness what is actually observed. As, 

 however, neither the formula of Bessel, nor that of Ivory, 

 very readily admits such modifications, Dr. R. used the 

 method given by the late Bishop of Cloyne, in the twelfth 

 volume of the Royal Irish Academy's Transactions, which, 

 however, he has extended to 85° zenith distance. 



If the atmosphere be supposed of uniform temperature 

 the refraction has been computed by Kramp ; it is found 

 greater than the truth. If the density be supposed to de- 

 crease uniformly as the height above the surface increases, 

 the refraction is given by Simson ; it is nearly as much in 

 defect as the other in excess, and it is found that their mean 

 is very near the truth. If then the differential equation of 

 refraction be developed in terms of the tangent of the appa- 

 rent zenith distance, it is found, on integrating, that the first 

 term belongs to an atmosphere bounded by parallel planes ; 

 the second depends on the equilibrium of the strata, 

 and the others alone are affected by the assumed hypo- 

 theses. Their geometrical means are found to satisfy the 

 Armagh observations as far as 85° zenith distance, below 

 which the series ceases to converge, and the mean changes 

 its relation to the true refraction according to the tempera- 

 ture and pressure. The expression thus obtained for re- 



