182 MR. IVORY ON THE THEORY OP ASTRONOMICAL REFRACTIONS. 



the results will be found to agree almost exactly with the short table inserted by 

 M. BiOT in the additions to the Con. des Temps for 1839, p. 105, the greatest differ- 

 ence between the computed quantities and the numbers in Halle y's table being 

 about 2". 



But this gives no intimation with respect to the particular constitution of the atmo- 

 sphere assumed in the calculation of the table. What is peculiar to a table in this 

 respect has no sensible influence on the refractions it contains except at altitudes 

 less than 16°. No definitive opinion can therefore be formed on the question, whether 

 Halley's table is the same which Newton computed and communicated to Flam- 

 steed on the principle that the densities are proportional to the pressures, without 

 comparing it with a sufficient number of refractions at low altitudes calculated from 

 the elements of Kramp. As the settling of this point may be thought not unimport- 

 ant, the following formula, which affords the means of computing the refractions at 

 all altitudes with exactness, has been investigated by reducing the integrals in the 

 expression of J ^ to serieses. 



} 



The series converges very slowly, which has made it necessary to continue it to ten 

 terms, the amount of which is still 3"-6 deficient from the exact quantity 2024"-2. 

 As the last terms decrease in the proportion of 2 to 1, it is obvious that the true sum 

 would be obtained by continuing the series : but the terms set down are more than 

 sufficient for the present purpose. 



