DETERMINATION OF ELLIPTIC ORBITS. 



123 



This gives 



('") 

 - 



From the corrected values of q lt ft, ft we may calculate a new residual , 

 and from that determine another correction for each of the quantities 



ft> ?2> ?8' 



It will sometimes be worth while to use formulae a little less simple 

 for the sake of a more rapid approximation. Instead of equation (19) 

 we may write, with a higher degree of accuracy, 



- = 'Ag, + @"A ?2 + @'"A ?3 + JZX A ?1 ) 2 + i2"(Ag 2 ) + jr"< Aft?, (21 ) 

 where 



dq 



Tlv -3 



(22) 



It is evident that " is generally many times greater than ' or X //x , 

 the factor B 2 , in the case of equal intervals, being exactly ten times as 

 great as A l B l or A 3 B 3 . This shows, in the first place, that the accurate 

 determination of Ag 2 is of the most importance for the subsequent 

 approximations. It also shows that we may attain nearly the same 

 accuracy in writing 



+ "A 



+ J"A 



(23) 



We may, however, often do a little better than this without using 

 a more complicated equation. For '+"' may be estimated very 

 roughly as equal to -J-X". Whenever, therefore, Aft and Aft are about 

 as large as Aft, as is often the case, it may be a little better to use the 

 coefficient ^ instead of in the last term. 



For Aft, then, we have the equation 



- C'"') = ('""O Aft + A( // @' // <5 / ) Aft 2 . (24) 



(5E"'" 7 ) is easily computed from the formula 



1 / n 2 

 '') = ( 1 5^, 



^2^ ^2 2 



.B. 



(25) 



which may be derived from equations (18) and (22). 



* These equations are obtained by taking the direct products of both members of the 

 preceding equation with @" x ('", @"' x ', and @' x @", respectively. See footnote 

 on page 119. 





