142 



DETERMINATION OF ELLIPTIC OEBITS. 



gve 



a = - -012595, 

 ! = - '04567, 



VI. 



Applying these corrections to the values of ft, q 2 , q s we compute 

 the second numerical columns under equations III^ III 2 , and III 3 . 

 We do not go on to the computations by equations III', etc., but 

 content ourselves with the old values of a l} b lt etc., G, and L, which 

 with the new residuals 



ft = '044949, y = -003012, 

 2 = '004952, 8 = - '04064. 



f 2 = 2 -X(% 2 + JC' 2 )Ag 2 = -004952 --02322(--77826 + -00247)Ag 2 . 

 This gives Ag 2 = '005042. 



As the term containing L has increased the value of Ag 2 by '00009, 

 we add this quantity to O l and (7 3 , and get 



Aft = - '04558, A^ 3 = - '04055. 



With these corrections we compute the third numerical columns 

 under equations III 1 , etc. This time we recompute the quantities a, 

 etc., with which we repeat the principal computations of IV, and get 

 the new values 



1= - -0167215 

 1= + -0149145 

 ?!=- 1576886 



a g =- -0335815 

 & 2 =- -0054413 

 c 2 =- -0779570 

 = 0090929. 



a 8 =- -0743299 

 & 8 =- -0098825 

 c 8 =- -0474318 



The quantities H and Z we neglect as of no consequence at this 

 stage of the approximation. 



With these values the new residuals 



gve 



a = + -0002919, ft= -'0000044, 

 Aft = C7 X = + -0010434, 



C7 



y= +'0000288, 

 + '0013222, 



= 3 =+ -0025316. 



These corrections furnish the basis for the fourth columns of 

 numbers under equations IIIj, etc., which give the residuals 



a = + -0000002, ft=+ '0000009, y = + '0000001 , 

 and the new corrections 



Aft= +'0000006, Ag 2 = +'0000021, Ag 3 = + -0000031. 

 The corrected values of ft, <? 2 , q s give 



log r, = 0-4282377, log r 2 = 0-4132937, log r 3 = 0-4061399. 

 We have carried the approximation farther than is necessary for 

 the following correction of the formula, in order to see exactly where 

 the uncorrected formula would lead us, and for the control afforded 

 by the fourth residuals. 



