THE OKJ3IT OF NEPTUNE. 67 



( , 



Putting 2, 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 be 



<fa r 



-7- cos (v /I), ... 



dv A (4) 



dh a 



a -= sm (v /I) ; 

 dr A 



and the coefficients of the equations of conditions will be 



(ft dh dv dh I dr 



.___ ,_ I fi ._.. 



dx dv de dr a de 



dh _ d?i dv eft I dr 

 dxf dv dn dr a dn 



dv\ d/l 1 /dr 



/(ft \ _ (ft /dv_\ (ft 1 /dr\ (5) 



\dh ) dv~ \dh) ~* a fo a \dh) 



d/l 1 /dr_\ 



a dr~ a\dk) 



The perturbations in the geocentric longitude of Neptune produced by Uranus 

 will be 



Jn 



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



* /7^ 



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



dr 



a dh 



n r ^Md^ 



Of course the effect 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 p the factor by which the assumed mass of Uranus must be 

 multiplied, so that the true mass shall be 



21000' 



the computed perturbations produced by Uranus will be the coefficients of fi 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 differently. 



