THE ORBIT OF URANUS. 



93 



Reduced Ej'pressions for the Latitude of Uranus. 



If we represent by T'i, T''^, F, the distances of Uranus from its descending nodes 

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

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

 disturbing planets. 



// 

 5/3 = — 0.0 11 4/ cos Fi 



— 0.0477<cos F2 



— 0.0125/ cos F 



_^ 0.245 



4- 0.386 sin ^ + 0.266 cos g 



— 0.043 sin 2^ -f 0-006 cos 2r/. 



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 ?« for 

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

 tion 



5^ = sin uh^ — sin ^ cos uhd 



u := 51 -(- Q -(- 2e sin g 



sin « = — e sin (j 



-)- cos (J sin g -{- sin u cos 7 

 -\- e cos id sin 2g -\- e sin u cos 2g 



