T11E OKB1T UF URANUS. 31 



CHAPTER II. 



APPLICATION OF TIIH I'KKiTMdM; MLTHOD TO THE COMPUTATION OF THE 



PLKTTKIIA TIO.V-. <n- i HANTS i;v SATTK.V 



Data of Computation. 



THE elements of Uranus, adopted in this computation, were deduced from the 

 comparison of nine normal heliocentric longitudes at intervals of 3697 days extend- 

 in- from 1781, December 26, to 1862, December 18, with corresponding provisional 

 places derived from the elements given in the "Investigation of the Orbit of Nep- 

 tune," with perturbations produced by Jupiter, Saturn, and Neptune. As the 

 perturbations are to be entirely re-computed, and the elements to be re-corrected 

 from more extended series of observations, all the details of this first approxima- 

 tion will be omitted. The resulting elements of Uranus are given in the follow- 

 ing table, together with the adopted elements of Saturn, which are nearly the same 

 as those employed in the theory of Neptune, except that the inclination and lon- 

 gitude of the node have been corrected to agree with observations: 



Elements II. of Uranus. Elements I. of Saturn. 



n 168 16' 31" 90 4' QT 



e 28 25 36.0 14 48 45.0 



9 73 11 58 112 20 



<p 46 20 2 29 39.2 



e .0469276 .0560050 



in seconds, 9679.5 11551.9 



n 15426.10 43996.13 



1 1 



m 



21000 3501.6 



In computing the perturbations of the radius vector, one of the largest terms 

 will be a constant. To avoid the necessity of computing separately the perturba- 

 tions of the second order, which depend on this constant, we shall include an 

 approximate value of it in the mean distance. This approximate value is, in the 

 action of an outer or an inner planet, <Hoga = m'MaW. #?'. In the action of 

 an inner or an outer planet, & log a' = + $ mM (J ( J> + a. D. b). M being the 

 modulus of the system of logarithms. 



Using the values of If and a D f b"[\ which are found in different works relating 

 to Celestial Mechanics, we find that the different planets produce the following 

 changes in 6 log a, the units being those of the seventh place of decimals: 



