26 ON THE PERTURBATIONS OF URANUS. [2 



32. I next entered upon a similar investigation, founded on the as- 

 sumption that the mean distance was about g^th part less than before, 



so that r or a = sin 31 = 0'515. The method employed was, in principle, 



06 



exactly the same as that given before ; but the numerical calculations were 

 somewhat shortened by a few alterations in the process, which had been 

 suggested by my previous solution. 



33. Assuming then that a = sin 31, the values of the quantities b, 



db ,d'b .,, 

 r , a" , - , will be 

 da da' 



77 Jt -J"1 



log & = 0-33385 log a = 9-57333 



log b, = 976106 logo^ = 9-86149 log a 2 ^ = 976573 



log 6., = 9-35361 log a = 971359 



da da 



= 8'98918 



Hence, by means of the formulae given before, the principal inequalities 

 of the mean longitude of Uranus, produced by the action of a planet whose 



m' 



mass is ----- , that of the sun being unity, and the eccentricity of whose 



e 



orbit is , may be found to be the following : 



20 



- 42-33 m' sin {nt - n't 4 e - e'} 

 + 76-55 m' sin 2 {nt - n't + e - e'} 

 + 7 -25 m' sin 3 {nt - n't + e - e 7 } 

 + 2-34 ?J7/ sin {n't + e'-ra-} 



4'74 mV sin {rit + d-Ts 1 } 

 + 41 '7 2m' sin {nt - Zn't + e - 2e' + CT} 

 -16-47 m'e' sin {nt - 2n't + e - 2e' + IT'} 

 + 33-93 m' sin {2nt - 3n't + 2e - 3e' + TO) 



- 63-41 m'e' sin {2nt - 3n't + 2e - 3e' + OT'}. 



