2] ON THE PERTURBATIONS OP URANUS. 33 



45. If, as before, we put e-e' = 0, and e-ra- = /3, it may be seen that 



] ,= - 4.2-33 sin -"-, = 76-55 sin 20 



m m 



A = _ 42 -33cos0 -^-, = 76-55 cos 20 



m m 



-^,= 7-25 sin 30 + 0-007460 ^ + 0-008974 % 

 m in m 



k *.= 7-25 cos 30 -0-008974 ^ + 0-007460% 



m m m 



-\= 0-20 sin( 0-/3)- 0-074738 - U-Jcos 20-%sin20l 

 m [TO m } 



% = - 0-20 cos ( - j8) + 0-074738 I 1 -, sin 20 + 7:i , cos 20} 

 m \m m 



V. y 



^;= 32-91 sin (20-/3) + 0-259765 j^; cos 0-^.sin 0} 

 m (lit m' } 



m 



%= 32-91 cos (20-^8) + 0-259765 \ P ] sin 0+^cos 0-. 

 w [TO //;. J 



46. Substituting these expressions in the above equations, and putting 

 for y8 its value, 50 15' 8, we obtain 



= - (1 -24872) sin + ( 1 '32231 ) cos - (1'48 1 1 0) sin 20 + (2'24265) cos 20 



- (1-48373) sin 30 + (2-22809) cos 30 + (9-26254)^- (9-50079) ?3 7 



+ (8-44376) I 1 , cos 0- q -. sin 0} - (8'02630) -[ ^ sin + % cos 01 

 v ' m, TO m m 



V XX J 



- (8-17031) (-^ cos 20- % sin 20} -(8'06861) I 1 -, sin 20+ % cos 20 . 

 v ' \m' m' J v ' [TO' TO' J 



0= (1-65190) sin + (2-06584) cos + (2-30220) sin 20 -(2*60306) cos 20 

 -(2-19916) sin 30- (2-15032) cos 30- (0-14305) &- (9'60933) ^, 



+ (9-34981) |cos0-sin 



-(8-85046) fe cos 20 -^-8^201 -(9-1 1828) jj sin 20 + ^cos20| 



