54 



H. JACOBT, 



Other motions of the telescope, such as would be caused by an azi- 

 muthal twisting of its mounting, are theoretically determinable; and motions 

 of this kind are precisely the ones most to be feared. They are also the 

 ones against which we are defenceless, when we employ the usual method 

 of fixing the position of the pole with the meridian circle by means of observ- 

 ations of the same star above and below the pole. Before proceeding, how- 

 ever, to a consideration of these errors, we shall deduce formulas which may 

 be used on the assuption that the telescope remained absolutely unmoved. 



We have n pairs of equations of the form (10) for each star. Let us 

 combine them into two mean equations of the form: 



««', -*-P, d o £ -*- sin o B n)S d\ — cos B njS dn -+- f ^ = 



. . (13) 



where &% and dr\ have lost the subscript n because they will be constants 

 on the supposition of an immovable telescope. 



If we now subtract these mean equations from each of the n others, 



we get: 



Ps A « J -+- A « Sin B n,s <*€ — \ COS B dfl H- A n ? = 



#A W — A n COsB n,s d ^ — \ Sm ^,^-*- A A,, = <> 



. . (14). 



These equations can be solved by least squares 3 ), and will make 

 known the values of: 



These may be substituted in the mean equations (13), and will give 

 us the values of: 



u ' s -?-Ps d o A 'i and v s -*-P s d o w - 



Knowing u' a -*-p s d A\ we can calculate u-*-d A by means of the 



equation: 



M < +d i= , ' i+ V;5 . . . (15). 



* i>s w sm 1" v 



With the help of equations (8), the quantities: 



u s -+- d A and v s -*- p s d w 

 now furnish a series of polar right ascensions which all differ from the 



2) The equations are not rigorously independent, as they should be if we are to solve 

 them by the method of least squares. This is evident from the fact that the elimination of 

 **'«"*" Ps rf o A aad v s -*-Ps d o u ought to have diminished the number of pairs of equations by 

 unity. The solution by least squares, however, will be practically satisfactory. 



M 



