THE ORBIT OF URANUS. 



93 



Reduced Expressions for the Latitude of Uranus. 



If wo represent by F,, F 2 , F, the distances of Uranus from its descending nodes 

 on the respective orbits of Jupiter, Saturn, and Uranus, we find the following 

 perturbations of the latitude, which are independent of the mean longitude of the 

 disturbing planets. 



it 

 &ft= 0.01 14< cos F, 



0.0477/ cos F, 



0.01 25< cos F, 



+ 0.245 



-j- 0.386 sin g + 0.266 cos g 

 - 0.043 sin '2g + O.OOG cos 2g. 



To find how far the last five terms may be represented by simple corrections to 

 the elliptic elements, we first represent the effect of minute corrections to the 

 inclination and node of Uranus as a function of its mean anomaly. Putting u for 

 the argument of latitude of Uranus, we have to a sufficient degree of approxima- 

 tion 



$(3 = sin utfy 

 = g -f- " + 



sin = e sin u 



sin <J> cos u&O 

 e sin g 



-|- cos o sin g -\- sin t> cos g 

 -j- e cos to sin 2g -4- e sin u cos 2g 



