Mr. Ivory on the Theory of the Astronomical Refractions. 347 

 fractions at low altitudes calculated from the elements of 

 Kramp. As the settUng of this pomtniay be thought not 

 unimportant, 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 expres- 

 sion of 8 9 to serieses. 



Tan $ = l^, e = tan f. 

 cos 9 ^ 



Log tan ^ = log secant 9 + 19-2133569 - 20. 



° f- " log. 



89 = sin9 X { e X 660-793 2-8200669 



+ e3 X 551-634 2-7337059 



+ e> X 371-268 2-5696873 



+ e^ X 219-762 2-3419630 



+ ^ X 116-763 2-0673034 



+ giix 58-170 1-7646976 



28-275 1-4514092 



13-797 1-1397974 



6-806 0-8329041 



3-311 0-5199046} 



e'^y. 



^15 ^ 



+ 

 + 



+ e^'x 

 + e'^x 



The series converges very slowly, which has made it necessary 

 to continue it to Ten terms, the amount of which i 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: bu 

 the ter^s set down are more than sufficient for the present 



^The'exact refractions calculated by the formula are next 

 to be compared with the numbers in Halley s table. 



Apparent zenith- 

 dist. 



20 

 40 

 60 

 70 



80 

 82 

 84 

 86 

 87 

 88 

 89 



Computed. 



Halley's Table. 



19-6 

 45-2 



1 33-1 



2 270 

 4 55-2 



6 3-9 



7 45-0 

 10 531 

 13 22-5 

 17 4-6 

 23 3-5 



20 

 45 



1 32 



2 26 

 4 52 



6 00 



7 45 

 10 48 

 13 20 

 17 8 

 23 7 



-0-4 

 + 0-2 

 + 1-1 

 + 1-0 

 + 3-2 

 + 3-9 

 0-0 

 + 5-1 

 + 2-5 

 -3-4 

 -3-5 



The examination that has now been made fully establishes 



