of the Orbits of Double Stars. 283 



k {t,-t,) = N.cotang?, |^ 4' (y + «) + (0 2 4) 

 k{t,-t,) = N cotang^|^^^(/3 + a) + (034) 



in all which, if the numbers of the tables are made use of, the 

 angles /3 — a &c. are to be expressed in minutes of an arc. 



In applying them the process will be nearly as follows. 

 The observed angles p &c. give a preliminary knowledge of 

 the values of a, /3, y, which will be sufficient for taking the func- 

 tions 4/ (|3— «) &c. from the table so accurately that their sub- 

 sequent alterations will never be considerable, their values 

 for an interval of 90° being confined between ^ and \. 



Making next, 



Ntang? , ,. . (0 12) _ , 



.o.^ N.tang?!,, , (0 13)_ , 



(21 -\ pi4'(y-«) = ci J -f-ZT = ^i* 



N^tangCa,, o^ , (0^3) _ , „ 



we obtain equations of this form: 



^ = C2 -X^^ + d^ &c. 

 ■^ sm 2 a 



In the first approximation the quantities c, &c. are consi- 

 dered as accurately determined, and that value of « is sought, 

 which, with its corresponding /3 and y, will satisfy two equa- 

 tions. Trials will soon effect this purpose. By means of these 

 values the quantities care corrected, and they will then be 

 obtained with an accuracy which hardly requires any further 

 correction. The new values of a, /3, y, thence deduced are 

 now substituted in the third independentequation, with the view 

 of ascertaining whether all observations belong to the same 

 ellipsis. Small differences are then corrected by an alteration 

 of the times. In the cases of greater differences the distances 

 for the less accurate observations must commonly be changed, 

 and then, indeed, the ??,?,, and the areas of the triangles on 

 which they depend, must be found by a new calculation. 



The formulae (A), (13), (C), (6), (8), and (20), which only 

 2 () 2 arc 



