80 



THE ORBIT OF URANUS 



If we subduct the effect in question when necessary, the remainder will be the 

 effect of the secular variation of the longitude of the perihelion of Uranus, to 

 which we shall revert presently. 



Let us next introduce such a change in the eccentricity of Uranus as shall pro- 

 duce the term 34".899 sing, and ascertain its effect on the other terms. For this 

 purpose we must determine le by the condition 



(2- e 2 )e = 



which gives 



e=17".464 = . 0000847. 

 A chane of this amount in le will introduce the followin terms in &v and 



cos 



= 34".899 sin g + 2".048 sin 2g 

 = 20 844 cos g 59 cos 2g. 



Subtracting these terms from the expressions previously found we have 



to= 0".144 sin 2g G".6S2 cosg 0".261 cos 2g. 

 cos 4fy = -j- 451 1 -f 167 cos g + 10 cos 2g 16 sin g -\- 1 sin 2g. 



Again, let us put 



efa = 3".342 == .0000162, 

 we shall have the elliptic terms 



&; = 6".682 cos g 0".391 cos 2g 

 cos 4-fy) = 1 62 sin g 11 sin 2g. 



Subtracting these expressions the constant terms, independent of the mean longi- 

 tude of the disturbing planets, are reduced to 



to= 0".144 sin 2j + 0".130 cos 2g. 



cos 4-fy> = 4511 -f 167 cos g -\- 10 cos 2g -\- 146 sin 5- -f 12 sin 2^. 

 0.43429 fy> = 1969 -j- 73 cos g -f 4 cos 2g + 63 sin g + 5 sin 2g. 



In the last equation we have introduced the constant -)-. 0000008 produced in $p 

 by the combined action of Venus, the Earth, and Mars. The effect of each planet 

 is computed by the approximate formula 



Secular Variations. 



The following inequalities result from the secular variations of the eccentricity 

 and longitude of perihelion produced by each of the disturbing planets, T being 

 the time expressed in centuries. 



From the variation of the eccentricity 



Action of Jupiter, 

 Action of Saturn, 

 Action of Neptune, 



* 1.216 T sin g 

 9.1 82 T sin g 

 0.502 T sin g 



0.072 T sin 2g> 

 0.5382' sin 2g 

 0.030 T sin 2g 



0. 005 T sin 3g 

 0.032 T sin 3g 

 0.002 T sin 3g 



