356 M. Bessel o?i a neiu Method qfjinding 



d(p = di . -^*^y- + ^ « . i sin 2 ^ . tan ^ {t' + t) 



J sin 2 ^ . sin t' o, r sin 2 ^ . sin t 



1 (cos t' + COS t) ' 2 (cos t! ■\- cos <) 



Let us now suppose that the instrument is correct to about 

 a minute, moving from east to west; it is evident that the co- 

 sine may be placed = 1, in the numerator of the result (tan (f) 

 and we shall obtain 

 tan <p = tan S . sec ^ (T' + t' — T — t) 



dm=zdl. 4^ + i idr'- dr) sin 2 ^ . tan ^ (T' + t'-T-t) 



r sin 23^^ ' - ^ 



Whence it appears that an error in the difference of the 

 correction of the time as shown by the clock (the influence of 

 which is so much smaller, the smaller T'— T is, in the case 

 in which this method is to be applied alone, viz. when the star 

 culminates near the zenith) may be considered as trifling. Such 

 an error would arise from the supposed incorrectness of the 

 clock; but we may suppose that this is generally much better 

 known than may be required in this method. 



There remains, therefore, only the error of declination, 

 whichis d<p = dl. ''"^f 



^ sm 2 5 



For a star passing thi'ough the zenith, the altitude of the pole 

 has exactly the same error as that of declination. For a star 

 culminating south of the zenith, the error is greater ; at least 

 in our latitudes. Suppose this method were to be applied to 

 determine the difference of latitude between two places, such 

 as 51° and 56°, and we were to select a star passing thi'ough 

 the southernmost point of the zenith ; the error in this point 

 would be = dl; and in the northern place =1'055 (i8. And 

 the error in the difference of the polar altitudes = 0'055 d S ; 

 or even for cZS = 2", only 0"*11. If both places were equi- 

 distant north and south from the 'tSth degree of latitude, the 

 difference would be found strictly correct. Moreover, it is 

 doubtful whether the absolute values of the divisions of the 

 zenith sector can be so correctly determined, that it may not 

 have, in an arch of 5°, a greater inaccuracy than 0"*11 ; at 

 least I consider this as much more difficult than the determi- 

 nation of the declination of a star to 2". 



This method is peculiarly recommendable on account of its 

 independence of any error in the instrument. If the collimation 

 should not be sufficiently corrected, the cylinders of the axis 

 should be unequal in their diameter, the telescope or the axis 

 should bend, &c., we shall still obtain a correct result, either by 

 reversing the axis between the two operations, or by observing 

 one day in one position and the next in the other position of 



the 



