T ii i: o K i: i T o i r u A .\ r s. 



Differentiating with resect to ^, we find for tin- terms of the second order 



,'//' , , 



6 ;//a/i/.-, sin 



. . cos 



Substituting for i-ATits numerical value just given and integrating, we find 



.Vi = mark, \ 2.S9.V sin A' (<>t>58* - I .-j(i/) cos N+ 1815" sin 2N\ 



- 3630"//i'uAv(< + constant ; 



,V. = w'arA-i { 2895* cos tf-f (6658" 4".260siu.y+ 1815" cos 2N\ 



-4- constant; 



the- constants being so chosen that fih and }k sliall vanish at the epoch. 

 lli-ilucing the values of M and Ik to numbers, they become 



rVi 5'.83 >iu ^+ (13".40 (r.867 7 ) cos JV 3".65 sin 2JV+ I'.OST 1 r.67, 

 ,V.- = 5 .82 cos N (13 .40 .86 T) sin N 3 .65 cos 2.V + 12 .12, 



the last two terms being arbitrary constants. 



Computing the values of these terms of M, < 7j, and /r, for intervals of 50 years, 

 from !()()() to 2000, we find them to be as follows: 



Y.-ar H }h Ik 



1600 -1".34 +0".10 -0*.02 



1650 -0.71 +0.05 0.02 



1700 -0.31 -1-0.02 -0.01 



1750 -0.10 0.00 0.01 



1800 -0.01 0.00 0.00 



1850 .00 .00 .00 



1900 0.00 0.00 0.00 



1950 -fO.04 0.00 -0.01 



2000 -f-- 18 -0.01 -0.02 



We see that although the ultimate effect of these terms is very considerable, 

 their effect, during the period that Uranus has been observed, is insignificant. 



Concluded Elements and Perturbations of Uranus. 



The corrections found in the last chapter being applied to the final provisional 

 elements (p. 99) give the following elements for 1850, affected by the great 

 inequality produced by Neptune: 



Elements IV of Uranus. 



Epoch, 1850, Jan. 0, Greenwich mean noon. 

 TT, 168 15' 6'.7 



f , 28 25 17.05 



6, 73 14 8.0 



$, 46 20.54 



,, .04(59230 



