DETERMINATION OF ELLIPTIC ORBITS. 125 



But a little examination will show that the coefficients of Ag 2 2 in 

 these equations will not generally have very different values from the 

 coefficient of the same quantity in equation (24). We may therefore 

 write with sufficient accuracy 



A?i = [Aft] + Ag 2 --[A? 2 ], Ag 8 = [Ags] + A? 2 -[Ag 2 ], (28) 



where [A^J, [A(? 2 ], [A</ 3 ] denote the values obtained from equations (20). 

 In making successive corrections of the distances q lt q 2 , q 3 , it will 

 not be necessary to recalculate the values of @', @", @'", when these 

 have been calculated from fairly good values of q lt q 2 , q s . But when, 

 as is generally the case, the first assumption is only a rude guess, the 

 values of @', @", @'" should be recalculated after one or two corrections 

 of q v q 2 , q s . To get the best results when we do not recalculate ', 

 @", @'", we may proceed as follows : Let @', @", @"' denote the values 

 which have been calculated; Dq lt Dq 2 , Dg 3 , respectively, the sum of 

 the corrections of each of the quantities q ly q 2 , q s , which have been 

 made since the calculation of @', @", @'"; the residual after all the 

 corrections of q lt q 2 , <? 3 , which have been made; and Ag 1? Ag 2 , Aq s 

 the remaining corrections which we are seeking. We have, then, 

 very nearly 



''"" (29 ) 



The same considerations which we applied to equation (21) enable 

 us to simplify this equation also, and to write with a fair degree of 

 accuracy 



(30) 

 (31) 

 where 



r . , ("'") ('"o FA , ('-) 



^ol" ~ ''^ L^ ( /2J- '"'"' L^^/sJ- '"''' 



Correction of the Fundamental Equation. 



When we have thus determined, by the numerical solution of our 

 fundamental equation, approximate values of the three positions of 

 the body, it will always be possible to apply a small numerical 

 correction to the equation, so as to make it agree exactly with the 

 laws of elliptic motion in a fictitious case differing but little from 

 the actual. After such a correction the equation will evidently apply 

 to the actual case with a much higher degree of approximation. 



There is room for great diversity in the application of this principle. 

 The method which appears to the writer the most simple and direct is 

 the following, in which the correction of the intervals for aberration 



