THE ORBIT OF URANUS. 



Tto representing the perihelion of 1850 = 168 15' 6*.7. We then have for the 

 eccentricity and longitude of perihelion at any epoch 



- -in (n 7r ) = Ih 



e cos (n Ttg) = e -f- W*. 



In the ahove terms multiplied by the time we have included the secular varia- 

 tions produced Ity Jupiter and Saturn. If the perturbations of the elements due 

 to each particular planet are required, we have 



Action of Jupiter, 



M = + 5-.737 1 ; Ik = (T.GOSr. 



Action of Saturn, ' 



lh = + y.5GT; Ik = 4.58971 



Subtracting these from the above expressions all the remaining terms will be due 

 to the action of Neptune. The values of H and 6 are due entirely to the action 

 dt Neptune. 



l-'or the sake of rigor, we may suppose the perturbations produced by each 

 planet to be multiplied by a factor representing the number by which the adopted 

 ina-s of the planet must be multiplied to obtain the true mass. 



It will add to the homogencousness of the theory to express the perturbations 

 of long period, which are multiplied by the product of the masses of Jupiter and 

 Saturn, as perturbations of the elements. These terms, as found on page 88, are 



(r.c.0) = 0".55 sin N< - 0".03 cos Jv% 



-f 40 .65 sin #, 10 .50 cos N, 

 (..!) = + 2. 64 sin N t + 4. 64 cos JV, 



+ 7 .35 sin N, + 4.41cosJV T 

 (r.c.l) = - 4 .23 sin N s - 3.87cosJv% 



+ 8.06sinJV T 8.38cosJV T 



These terms, together with the arbitrary corrections of the elements which have 

 been applied to make them very small at the epoch, may be replaced by the follow- 

 ing corrections to the elements: 



II = 0*.55 sin JV 6 0".03 cos N t 



+ 40 .65 sin JV T 10 .50 cos JV T 



+ 27".27--ll".727'. 



& =+ 2 .09 sin N t + 1 .94 cos N< 



2. 13 sin #,+ 3.71cosJv* T 



+ 1".28. 



& = + 1 .32 sin N s + 2 .32 cos N t 

 -f 3 .68 sin N, -j- 2 .21 cos N, 



&o = 27 sin JV 7 -f- 104 cos JV 7 + 76 (in units of the 7th decimal). 

 The amount of the perturbations of the elements for every half century, from 

 the year 1000 to 2200, is given in the following table. Column (1) gives the per- 

 turbations by Neptune, Saturn, and Jupiter, computed from the expressions 



