34 ON THE PERTURBATIONS OF URANUS. [2 



= (1-91407) sin 6- (2-55189) cos 6- (2'62790) sin 20- (2'64230) cos 20 



-(2-25331) sin 30-(2'34185) cos 30 + (9'96344) j+ (0'56029)^, 



- (9-83835) te cos - % sin 01 + (9*64968) fe sin + % cos 01 



' \m! m! J ' \m m! } 



+ (9-45371) -fe cos 20 - % sin 201 + (9'47306) fe sin 20 + % cos 201 , 

 v ; [TO TO J [TO t J 



where the numbers enclosed within parentheses denote the logarithms of 

 the corresponding coefficients, as before. 



47. From these equations we find, by the same method as before, 

 0=-4655' ;=138"'92 %=-109"'83 



m m 



Hence, since e = 21755', e' = 26450 / , the mean longitude of the disturbing 

 planet at the epoch 1810"328. The sidereal motion in 3G synodic periods 

 of Uranus = 57 42', Precession = 30'. .'. mean longitude at the time 1846"762, 

 or October 6, 1846, =323 2'. 



Also, the expressions for . and . are 



m m 



* = 33"-93 sin (30.-^) - G3''41 e' sin (30 - ft') 



II V 



- = 33-93 cos (30 -^8) -63-41 e'cos(3e-ft') ; 



it v 



w 



here e rs' = }'. 



Equating these to the values given above, we find e' = 2'4123, /8' = 279 14', 

 and .-, CT' = 29841'. Hence longitude of the perihelion in 1846 =299 11'. 



Lastly, substituting the values just obtained in equation (l) of Article 

 39, we find m' = 075017. 



48. Hence the values of the mass and elements of the orbit of the 

 disturbing planet, resulting from the second hypothesis as to the mean 

 distance, are the following : 



-, = 0-515 

 a' 



