410 On the Calculation of the Orbits of Double Stars. 



quantity a b 9 which is easily calculated if once an hypothesis 

 respecting one of the quantities a, /3, y has been adopted, would 

 be considered as unknown. It would be supposed that all the 

 four points belong exactly to the same ellipse, which is de- 

 scribed agreeably to the Keplerian law ; and as this supposi- 

 tion is never perfectly true, we should be obliged to begin a 

 useless repetition without having a guide how to correct the 

 error, if the value of a b obtained by substitution in the three 

 equations is not found to agree with its value derived from 

 equations (15). 



It is much better for the purpose to select those two of 

 the equations (16) which belong to the best determined points, 

 to assume by hypothesis one of the quantities, for instance 

 a ; with this quantity the values of |3, y and a b are to be 

 calculated by means of the equations (8) and (15), and the 

 value of k is then to be deduced from each of the two equa- 

 tions (16). The value of a is to be changed until the values of 

 k agree, and the quantities thus obtained are then to be sub- 

 stituted in the third equation. If little is wanting of the iden- 

 tity of the values, one may make a trifling change in the mo- 

 ments of time until everything agrees. Our observations will 

 not, for a long time to come, be so accurate as not to admit 

 of an alteration of a month, or O'l year. If the deviations are 

 great, we may change for the less accurate observation either 

 the distance, or the angle of position, which, indeed, renders 

 necessary an entirely new calculation of ?, 19 2 , and a com- 

 plete repetition. The less accurate observation being, how- 

 ever, entirely contained in one of the equations (16), a little 

 reflection will, in most cases, easily show the sign and the 

 approximate value of the correction which is to be applied. 



For facilitating these trials a table for the function 



(17) $(x) = 2 x sin 2 x 



has been appended to this article, giving the value of the func- 

 tion for every ten minutes from x = to x = 90. The 

 angle x being always equal to half the difference of two ex- 

 centric anomalies, the table will immediately serve for any 

 two observations, whose interval does not exceed half a revo- 

 lution, which, as such may be selected from the equations (16), 

 will generally not be the case. For other cases the values 

 may be easily calculated, as 



<p (x} = 2?r $(180 x). 



The table has been calculated to seven decimal places, but 

 five only have been given, as these will always be sufficient. 



[To be continued.] 



LXII. Ob- 



