40 



THE ORBIT OF NEPTUNE. 



dl dh dk ' 



h log r z= 5 log a + — ^ bh + -^^ dk, 



dh 



dk 



^^=P^+f,^- 



For the period during which Neptune has been observed, we have, to a sufficient 

 degree of approximation, 



dv 





dl 



= 1, 



dv „ , 

 -Tfj- = — 2 cos ?, 

 dh 





dv o • 7 

 -=rr ^ Zsva. I ; 

 dk 



— #- = — IfsmZ, 

 dh 





d log r 



■ — j~ rz — M COS I; 

 dk 



d(3 



-^ =: — COS V, 

 dp 





-i— =z sm V. 

 dq 



The values of P^.i, P^.u etc., derived from the perturbations by Uranus, are, putting 

 A — mean longitude of Uranus, minus that of Neptune, 



P,.i = 



— 0".683 s; 



— .400 s 

 + .044 s; 

 + .006 s 

 + .009 s 

 + .001 s 



— .00.3 s: 



— .002 s: 



— .002 s 



— .001 s: 



n A 



n 2A 



u 3.4 



n 4^ 



n 5 ^ 



n 6^ 



n lA 



n 8^ 



n 9^1 

 nlO.4 



P>.'.= 



— 0".021 



— .2-54 cos A 



— .867 cos 2 A 



— 1 .821 cos 3 A 

 + .355 cos 4.4 

 + .083 cos 5 A 

 + .039 cos 6^ 

 + .018 cos 7 A 

 + .008 cos 8^ 

 + .004 cos 9 A 



B...I = — 23 



5".000 cos A 



11 .410 cos 2 A 



2 .031 cos 3 A 



.462 cos iA 



.165 cos 5 A 



.076 cos 6 A 



.035 cos 7 A 



.017 cos 8^ 



.008 cos 9 A 

 .004 cos 10 A 



— 0".038 sin A 



— .035 sin 2 A 



— .147 sin 3 .4 

 + .023 sin 4 A 

 + .006 sin 5 A 



+ 4 cos 2 ^ 



J?,i = + 0".328 cos A 

 + .005 cos 2 A 



— .078 cos 3 A 



— .022 cos 4 A 



— .009 cos 5 A 



— .006 cos 6 A 



— 58 sin A 



— 66 sin 2 A 

 + 34 sin 3 ^ 

 + 7 sin 4^ 

 + 3 sin 5 A 

 + 2 sin 6 ^ 



+ 0M16 sin A 

 + .048 sin 2 A 



— .017 sin 3 A 



— .003 sin 4 A 



1 = + 4". 208 sin A 



+ 10 .892 sin 2 A 



— 1 .989 sin 3 A 



— .418 sin 4^ 



— .135 sin bA 



— .056 sin 6 A 



— .023 sin 7 A 



— .009 sin 8 A 



— .004 sin 9 A 



— .002 sin 10 A 



= — 0".014 



— .022 cos A 



— .027 cos 2 A 



— .135 cos 3 4 

 + .025 cos 4 A 

 -|- .008 cos 5 A 

 -f .002 cos 6 A 



S.1 = - 7 



— 46 cos A 



— 70 cos 2 A 

 + 32 cos 3 A 

 + 9 cos 4^ 

 + 3 cos 5 A 

 + 2 cos 6 ^ 



5„j = + 0".148 cos A 

 + .002 cos 2 A 



— .035 cos 3 4 



— .009 cos 4 A 



— .004 cos 54 



— 0".559 cos A 



— .330 cos 2^ 

 + .078 cos 3 .4 

 + .018 cos 4 4 

 -I- .013 cos 5 4 

 + .003 cos 6 A 



+ 0" 

 + 0. 



+ 1- 



— . 



— 



— . 



— , 



— . 



— 



228 s 

 .863 

 .849 s: 

 .355 s: 



085 s 

 .041 s: 

 .018 s: 

 .008 s 

 .004 s: 



n A 

 n24 

 na4 

 n 4.4 

 n 5 4 

 n64 

 n74 

 n 84 

 n94 



— 1 sin 3 4 

 + 2 sin 4 4 



— 0".256 sin A 



— .105 sin 2 4 

 + .036 sin 3 4 

 + .008 sin 4 A 



