40 



THE ORBIT OF NEPTUNE. 



dv . dv . dv 



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



degree of approximation, 



dv 



dl 



, 



-jj- 2 COS /, 



an, 



d log r 



- 

 dh 



-f- =. COS V, 



dp 



= 1, 



dv 0-7 



-jj- = 2 sm Z: 

 dk 



d log r _ 

 ~~dk~~~ 



d{3 



-j = sin v. 



dq 



Theyalues of P s _i, P c , t , etc., derived from the perturbations by Uranus, are, putting 

 A =: mean longitude of Uranus, minus that of Neptune, 



r. 3 = o".02i 



.254 cos A 



.867 cos 2 A 



I .821 cos 3 A 

 + .355 cos 4 A 

 4- .083 cos 5 A 

 4- .039 cos 6 A 

 4- .018 cos 7 A 

 + .008 cos 8 A 

 + .004 cos 9 A 



R^ = 23 



4- 4 cos 2 A 



0".038 sin A 



.035 sin 2 A 



.147 sin 3 A 

 4- .023 sin 4 4 

 4- .006 sin 5 A 



,i = 4- 0".328 cos A 

 4- .005 cos 2 4 



.078 cos 3 A 



.022 cos 4 X 



.009 cos 5 A 



.006 cos 6 A 



58 sin A 



66 sin 2 A 

 4- 34 sin 3 A 

 4- 7 sin 4.4 

 4- 3 sin 5 A 

 4- 2 sin 6 A 



4- 0".116 sin .4 

 4- .048 sin 2 A 



.017 sin 3 A 



.003 sin 4 A 



.002 sin 10 A 



0".014 



.022 cos A 



.027 cos 2 A 



.135 cos 34 

 4- .025 cos 4 .4 

 4- .008 cos 5 A 

 4- .002 cos G .4 



46 cos A 



70 cos 2 A 

 4- 32 cos 3 A 

 4- 9 cos 4 A 

 4- 3 cos 5 A 

 4- 2 cos 6 A 



4- 0".148 cos A 

 4- .002 cos 2 ^ 



.035 cos 3^4 



.009 cos 4 A 



.004 cos 5 .4 



4- 0".228 sin A 

 4- .803 sin 2 4 

 4- 1 .849 sin 3 A 



.355 sin 4 A 



085 sin 5 A 



.041 sin 6 A 



.018 sin 1 A 



.008 sin 8 A 



.004 sin 9 A 



1 sin 3 A 



4- 2 sin 4 4 



0".256 sin A 



.105 sin 2 A 

 4- .036 sin 3 A 

 4- .008 sin 4 A 



