94 THE ORBIT OF URANUS. 



COS w = — e cos (J 



-|- cos (J cos g — sin (J sin g 

 -\- e cos w cos 2g — e sin tj sin 2g. 



Substituting these values of sin u and cos u in the expression for h[3, and putting 

 sin ^hB = b'd, we have 



h^ := e cos ah'O — e sin G) h^ 



-\- ( cos Ldh^ + s"'^ "'^'^0 ™^ ff 4~ ( ^i'^ ""^^^ — '^'^^ "f^'^) ^^^ 9 

 -\- (e cos aih<p -\- e sin ofVO) sin 2ij -j- (e sin wr)^ — e cos (0^0) cos 2//. 



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



cos u>h^ + sin (OfVO = 0".386 

 sin (jh(p — cos o^'O = .266. 



Since o =: 95° 3', this gives 



I! 



h^ = 0.231 ; 

 h'd^ 0.409; 



h[3 = — 0.013 



+ 0.3S6sin (7 + 0.266 cos ^ 

 + 0.018 sin 2(/ + 0.013 cos 2r/ 



Subtracting this expression from the corresponding terms of 5/3, we have left 



5/3 = + 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 



r, = v — 127° 37' 

 F2 = v — 126 45 

 V^ = v — 155 32 



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

 term by the factor (1 -(-,«) by which the adopted mass of Neptune, yy^oo' '""^* 

 be multiplied to obtain the true mass, we find 



5/3 = (4".69 4- 1".14«) Tcos v — (5".24 + 0".52//)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 " Tubles 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. 



