2] ON THE PERTURBATIONS OF URANUS. 13 



f3 (i_- l)(2t -1) ' 1 ft-- 



\2 (z - ) 2 z"(z - 2w) + 2 " z (z - 2n) J '-' 



f3 (t'-l)n' ,1(1-1)"' _ 



a 2z(z-2n) z(z- 



z (z n) (z In) du- 



a 



ll. Now, if we assume > or a = sin 30 = 0'5, the values of the funda- 

 db d*b 



mental quantities b, a -,- , a" -,- , will be 



77 7-7 



log 6,, = 0-33170 loga ( (7a = 9'53765 log a 2 rf - -977848 



77 7-7 



log 6, =974497 log a'- 1 =9-83868 loga 2 -j- =9'70857 



log b, =9-32425 loga ,-' = 9'68012 loga 2 . T 2 = 9 "87776 



( ' CC ( * Ct 



log b 3 = 8-94670 loga -,- l= 9'46315 loga 2 C ~ = 9'86253 



da da~ 



Hence the principal inequalities of mean longitude, produced by the 

 action of a planet whose mass is _ -- , that of the Sun being unity, and 



UUU 



the eccentricity of whose orbit is will be the following: 



- 36-99 nt sin {nt-nt + e-e} 

 + 58-97 TO' sin 2 {nt - n't + e - e'} 

 + 5'80 TO' sin 3{nt-n't + - e'} 

 + 2 '06 m' sin [n't + f. 1 -&} 



- 4-30 me' sin [n't + e' - is'} 



+ 31 -25 TO' sin {n-2;iV + e-2e' + ra-} 



-12-14 m'e' sin {nt - 2n't + e - 2e' + w'} 



+ 48-55 TO' sin {2nt - 3n't + 2e - 3^ + ra-} 

 -93-01 m'e'sin 



