94 THEORBITOFURANUS. 



cos u = e cos o 



-(- cos a) cos <7 sin cj sin <? 

 -|- e cos u cos 2g e sin co sin 1g. 



Substituting these values of sin u and cos u in the expression for 5/3, and putting 

 sin $50 = 5'0, we have 



5/2 = e cos 6>5'0 e sin co <<> 



-|- ( cos 6)5$ -f- sin o>5'0) sin g -\- ( sin G>5</> cos o5'0) cos # 

 -J- (e cos 6>5<> -j- e sin o5'0) sin 2</ -(- (e sin (j<5< e cos u5'0) cos 2jf. 



To represent the numerical coefficients of sin g and cos g in 5/3 we must put 



cos G)5$> + sin o5'0 = 0".386 

 sin 6)5$ cos o5'0 = .266. 



Since o = 95 3', this gives 



// 



5<2> = 0.231; 

 i'0 = 0.409; 



S^ = 0.013 



// 



+ 0.386 sin (7 + 0.266 cos # 

 -j- 0.018 sin 2^ + 0.013 cos 2g 



Subtracting this expression from the corresponding terms of 8(3, we have left 

 8/3 = + 0".258 0".061 sin 2g 0".007 cos 2g. 



The first term of this expression shows that the mean orbit of Uranus at the 

 present time is a small circle of the sphere one-quarter of a second north of its 

 parallel great circle. 

 If we put 



v = longitude of Uranus in its orbit, referred to the equinox and ecliptic of 

 1850, we have 



V l = v 127 37 

 F 2 = v 126 45 

 V 3 = v 155 32 



Substituting these values in the first three terms of 5/3, and multiplying the last 

 term by the factor (!+/*) by which the adopted mass of Neptune, TT TT , must 

 be multiplied to obtain the true mass, we find 



Sp = (4".69 + l".14a) 7'cos v - (5".24 -f 0".52 1 )7'sin v. 



To these terms must be added those which arise from the motion of the ecliptic. 



In the absence of any exhaustive investigation of the obliquity and motion of 

 the ecliptic, I adopt the elements of Hansen, employed in his " Tables du Soleil" 

 because they are a mean between the results of others, and are very accordant 

 with recent observations. The secular motion of the obliquity there employed is 



46".78. 



