122 



DETERMINATION OF ELLIPTIC ORBITS. 



distance of the body from the foot of the perpendicular from the sun 

 upon the line of sight. If we set 



equations (8) become 



Let us also set, for brevity, 



Then 



@ 



2 , 



(12) 



may be regarded as functions respectively of 



7 j 



dq l 



Pi> /2> /3> therefore of ft, <? 2 , ft, arid if we set 



and 



we shall have 



To determine the value of ', we get by differentiation 



'=^ 

 But by (11) 



Therefore 



(13) 

 (14) 



(15) 



(16) 

 (17) 



T a 



3 ' 



(18) 



Now if any values of q lt q 2 , q 3 (either assumed or obtained by a 

 previous approximation) give a certain residual @ (which would be 

 zero if the values of q lt q 2 , q s satisfied the fundamental equation), and 

 we wish to find the corrections Aq 1} Ag 2 , Ag 3 , which must be added 

 to q lf q 2 , q 3 to reduce the residual to zero, we may apply equation (15) 

 to these finite differences, and will have approximately, when these 

 differences are not very large, 



j + "Aft + '"Aft. (19) 



