A 



MR. IVORY ON THE THEORY OP ASTRONOMICAL REFRACTIONS. 181 



such as is mentioned in the correspondence between Newton and Flamsteed. The 

 formula for the refraction in the supposed constitution of the atmosphere has been 

 ^iven both by Kramp and Laplace ; and it may be taken from the Paper in the 

 Philosophical Transactions for 1823, p. 441, 



X = 4- = -19601, A = a/ cos2 ^ + 2 i s, 



+ (^- 2X2 + 2X3)./ 2 ''"-'' 



+ '-?■/ 



the integrations extending from s = to s = (^ . The coefficients of this formula 

 are as follows : 



A=l— X + -2---6= '82193 

 B = X — 2 X2 + 2 X3 = -13423 



C=4>^'--|5^' =-02377. 

 D = I" X3 = -02007. 



For the horizontal refraction, when cos ^ = 0, A = />/2 i s, we obtain by the usual 

 integrations, 



Jj^=:^X |A^-B^/T^-C^/3 + D^/T|: 



or in seconds, h d = 2024-2 instead of 2025" as in Halley's table. This proves the 

 exactness of Kramp's elements. 



With respect to the other numbers in the table a distinction must be made. In 

 every table of refractions, whatever be the constitution of the atmosphere on which 

 it is founded, the numbers answering to altitudes greater than 16°, depend only upon 

 one element, namely, the refractive power of the air. Reckoning from the zenith as 

 far as 74°, any table may be deduced from any other, provided both are accurately 

 calculated, merely by a proportion. In the table published annually in the Con. des 

 Temps, the refraction at 45° is 58"'2 : and, if Halley's table has been accurately 

 computed, the numbers in it, between the limits mentioned, will be equal to the like 



540 90 



numbers in the French table multiplied by ^, = g^. The calculation being made. 



