138 



DETERMINATION OF ELLIPTIC ORBITS. 



XII. 



When an approximate orbit is known in advance, we may use it to 

 improve our fundamental equation. The following appears to be the 

 most simple method : 



Find the excentric anomalies E I} E 2) E 3) and the heliocentric 

 distances r 1? r 2 , r 3 , which belong in the approximate orbit to the times 

 of observation corrected for aberration. 



Calculate S lt B 3> as in 1, using these corrected times. 



Determine A lt A 3 by the equation 



sin ( E 3 E 2 ) 



sin E 2 sin (E 2 E^) 



in connection with the relation A l -}-A 3 = 



Determine B 2 so as to make 



sn 



r 



' 1 



' 



4 sin J (E 2 - EJ sin | (E 3 - E 2 ) sin \ (E 3 - EJ 



equal to either member of the last equation. 



It is not necessary that the times for which E lt E 2 , E 3) T I} r 2 , r 3 , 

 are calculated should precisely agree with the times of observation 

 corrected for aberration. Let the former be represented by /, 2 ', t 3 , 

 and the latter by /', 2 ", t s "', and let 



We may find 



A log r 3 = log( tj' - C) ~ log( ^ - O 



5 3 , ^L 1} -4 3 , j5 2 , as above, using /, ^ 2 ', t s ', and then use 



AlogTj, AlogT 3 to correct their values, as in VIII. 



Numerical Example. 



To illustrate the numerical computations we have chosen the 

 following example, both on account of the large heliocentric motion, 

 and because Gauss and Oppolzer have treated the same data by their 

 different methods. 



The data are taken from the Theoria Motus, 159, viz., 



The positions of Ceres have been freed from the effects of parallax 

 and aberration. 



