2] ON THE PERTURBATIONS OF URANUS. 25 



30. These equations may be rapidly solved by approximation. The 



nf\ fy 



coefficients of " and , in the first equation being small, we may find 

 from it an approximate value of 6, the substitution of which in the second 



'29 Q 



and third equations will give approximate values of *-j and , . By means 



of these a more accurate value of 6 may be found from the first equation, 

 and the process being repeated, will enable us to satisfy all the equations 

 as nearly as we please. 



Thus we find 6= -51 30', & = 27l"'57, %= -207"'24. 



m' m' 



Now e is known and =217 55' .'. e' = 26925' the mean longitude of 

 the disturbing planet at the epoch 1810-328. The sidereal motion in 36 

 synodic periods of Uranus = 5 5 12', precession = 30', .'. mean longitude at the 

 time 1846762, or October 6, 1846, =325 7'. 



Also, the analytical expressions for . and --.- are 



J m m 



^ = 48"-55 sin (30 -/3)- 93-01 e' sin (30 -ft') 



m 



-2> = 48-55cos(30-)-93-01e'cos(30-/3') 

 m 



where e zr'^/3'. Equating these to the values given above, we find 



e' = 3'2206, /3' = 26228', and .'. n7'=31527'. 

 Hence longitude of perihelion in 1846 = 315 57'. 



Lastly, substituting the values just obtained in equation (l), we find 

 m' = 0'82816. 



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

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

 distance, are the following : 



-, = 0'5 

 a 



Mean Long, of the Planet, October 6, 1846 325 7 



Longitude of the Perihelion 31557 



Eccentricity of the Orbit 0'16103 



Mass (that of the Sun being 1) 0'0001656 



These are the results which I communicated to the Astronomer Royal 

 in October, 1845. 



A. 4 



