10 ON THE PERTURBATIONS OF URANUS. [2 



except in the equation depending on the mean longitude of Saturn minus 

 twice that of Uranus, the error of which had been already pointed out by 

 Bessel. The principal equation depending on the action of Jupiter also 

 required correction, in consequence of the increased value which has been 

 lately obtained for the mass of that planet. The corrections to be applied 

 to Bouvard's tables on these accounts are the following : 



+ 1 -9 1 8 sin {<k - 2< 2 - 1 3 1 '5} 

 + 1-085 sin {(-,} 



<, <,, <., being the mean longitudes of Jupiter, Saturn, and Uranus, respec- 

 tively. In the reduction of the Greenwich Observations, the latter correction 

 was already taken into account. M. Hansen having also found some new 

 inequalities in the motion of Uranus, depending on the square of the 

 disturbing force, I re-computed the values of these, following the same 

 method as that given by M. Delaunay in the Conn, des Temps for 1845, 

 and my results agreed very closely with his, the terms to be added to 

 the longitude being 



+ 32' 00 sin {3<k - G<, + 2< + 22 1 8 '8} 

 8 "35 sin {2^-6^ + 2<f> + 39 10'5| 



4 48'4}. 



With respect to the inequalities of higher orders neglected by Bouvard, 

 I considered that the most important of them would be, either those of 

 long period, or those whose period was nearly equal to that of Uranus. 

 During three-fourths of a revolution of the planet, the effects of the former 

 class would be nearly confounded with those arising from a change in the 

 epoch and mean motion, and those of the latter class with the effects 

 produced by a constant change in the eccentricity and longitude of the 

 perihelion. The position of the planet to be determined would, therefore, 

 be little affected by these terms, and the others would probably be much 

 smaller than those which would necessarily be neglected in a first approxi- 

 mation to the perturbations produced by the new planet. 



8. Taking into account the several corrections above-mentioned, the 

 residual differences between the theoretical and observed heliocentric longi- 

 tudes were the following : 



