of Bessel's Astronomical Observations. 255 



If the errors of division had not been taken into account, it 

 is evident from this comparison that the upper passages would 

 have given the polar distances sensibly greater in the eastern 

 position of the circle, than in the western position : with re- 

 gard to the lower passage the contrary would take place ; the 

 difference is indeed not very great, but I am confident that 

 this excellent instrument can determine still smaller quantities ; 

 and I consider the correction obtained by the investigation of 

 the errors of division, though it never exceeds 0" - 6, as an es- 

 sential addition to the accuracy of the declinations, the esta- 

 blishing of which is the object of this investigation. After 

 allowing for the errors of division and of flexure, the agree- 

 ment is as satisfactory as the accidental errors of observations 

 and the errors of the curve that undoubtedly still remain 

 would allow us to expect. 



The errors of division applied for the place of the pole 

 are taken from the table in Art. I. in the eastern position at 

 4-0"-21, in the western at -f 0"*32 : but it cannot be supposed 

 that the curve should give them quite correctly; and if they 

 are not correct, a constant difference of the polar distances 

 observed in both positions of the instrument will be produced, 

 which may be determined to advantage by the comparison of 

 several stars, the observations of which depend on diflerentparts 

 of the circle. Calling the true errors of division +0''*21-fx 

 and +0"'32 + .r', we have for the upper passages 



* + x'=East Polar Distance — West Polar Distance. 

 And for the lower passages 



x + x'= West Polar Distance — East Polar Distance. 

 By a mean of 100 comparisons I found 

 x + x'= — 0"-028. 

 The comparison of the upper passages with the lower ones 

 will give (in the next article) x—x', by which x and x' may be 

 determined in such a manner that if the places of the pole ob- 

 tained by the two pole-stars be thereby corrected, these may 

 be considered as having been derived from a mean of all stars 

 that have been observed. 



6. Refraction. 

 The polar distances observed in both passages, and cor- 

 rected for the change of the thermometrical factor, the errors 

 of division and flexure, as given in the preceding article, still 

 involve the error of the mean refraction. This f shall so de- 

 termine as to obtain the greatest possible agreement for both 

 passages. Let the refractions employed in the calculation for 

 the upper passage of a star be called g, for the lower one g', 



that 



