256 Introduction to the Seventh Sectiori 



that for the pole r : again, let the corresponding true refrac- 

 tions be denoted by q{l+k\ §'(l+£), rCl+k); and let the 

 polar distances calculated from the mean of all observations of 

 the upper passage be = P ; the same from the mean of the ob- 

 servations of the lower passage be = P', the true polar distance 

 is =V-Uz-x') + (r + §)k = ?' + l>{*-x') + {§'-r)k 



and hence the equations of condition for the refraction 

 = P'-F + (x-x') + (g'±§-2r)L 

 In order to give a proper value to these equations, and in 

 general more accurately to determine the capability of the in- 

 strument, I must begin with the investigation of the probable 

 errors with which the results of the circle, when corrected 

 by the table of the errors of division, are still affected. This 

 investigation may be grounded on the differences of the polar 

 distance in both positions of the circle ; if the probable con- 

 tingent error of observation be expressed by e, the probable 

 error of division, after applying the correction in the table by 

 e, the probable difference of the results of a eastern, and a 1 



western observations is = \/ \ 2<r + — + — ■ \ , and we have, 



l a a' 1 



if the really existing difference be called o, 



(0-6745) 2 £[u + 0"'028] 2 = 2ne 2 + 5:^+ £) 



where n signifies the number of polar distances that have been 

 compared. Assuming e as determined in the 3d article, all 

 the stars from the pole to a Cygni inclusive give 

 e= ± 0"*2794. 

 This e is the limit of the probable accuracy of a polar di- 

 stance observed in one position of the circle only, which by 

 ever so great a repetition of observations never can be further 

 diminished ; for a determination made in both positions of the 



circle the same limit is =— — -= +0"*1976. 



If a result be required which is to have a much smaller pro- 

 bable error, it can only be found by employing for its deter- 

 mination observations depending on distant points of the limb 

 of the circle : the nature of the instrument requires, therefore, 

 to have recourse to that means, wherever it is possible to do 

 so, for determining refraction, latitude, &c. 



The equations of condition for refraction have the probable 

 errors 



V Jt t 2 « Va'-ta/ ^ 2 C \b'+b/ \ a'+a ~ b+b> J 



where s', b, b' signify the same for the lower passage which e, 

 a, a' express for the upper one. From this formula the 

 numerical values of the errors which are contained in the fol- 

 lowing table have been computed. 



