THE ORBIT OF NEPTUNE. 67 



Putting /I for the geocentric longitude, and A for the distance from the earth, 

 the differential coefficients of the geocentric with respect to the heliocentric co- 

 ordinates will he 



d% _r I ^^ 



(4) 



' 



d% _ 



dv 



= — cos (v - 



A ^ 



-^) 



> 



a 



d% _ 

 dr 



- — sm [v - 



A ^ 



-^) 



} 



and the coefficients of the equations of conditions will be 



d7._ 

 dx 



d% 

 dv 



dv , dX 



1 



a 



dr 

 di 



d% _ 



dx'~ 



d?i 



dv 



dv d'k 

 dn dr 



1 

 a 



dr 

 dn 



\dli ) 



_fZZ 

 dv 



^ (:!)+» 



dx 

 dr 



l/dr\ 

 a \dh) 



\dk) 



_d?L 



~ dv 



^ (l)+« 



d% 



dr 



l/dr\ 



a\dk) 



(5) 



The perturbations in the geocentric longitude of Neptune produced by Uranus 



will be — 



d% 



1. Perturbations of the true heliocentric longitude multiplied by y— ; 



d'^ 



2. Perturbations of radius vector multiplied by -^, for which has been taken 



^^""^'^mfr 



Of course the eflFect of the long-period and secular perturbations of the elements 

 produced by the action of Uranus must be included in the perturbations of 

 Neptune. 



Representing by ^i the factor by which the assumed mass of Uranus must be 

 multiplied, so that the true mass shall be 



1 + F 

 21000' 



the computed perturbations produced by Uranus will be the coefficients of jtt in 

 the equations of condition. 



§ 30. The residuals in longitude thus give the following equations between the 

 unknown quantities, which are numbered in the order of time, but grouped 

 somewhat diffi^rently. 



