6 



THE ORBIT OF URANUS. 



CHAPTER I. 



METHOD OF DETERMINING THE PERTURBATIONS OF LONGITUDE, RADIUS 

 VECTOR, AND LATITUDE OF A PLANET BY DIRECT INTEGRATION. 



LET us conceive a plane determined by the condition that it shall pass through 

 the sun and contain the tangent to the orbit of a planet at any moment. If the 

 planet were acted on by the sun alone, the position of this plane would be invariable, 

 but, under the influence of the disturbing forces of the other planets, it is subject, 

 at each instant, to a motion of rotation around the radius vector of the planet. We 

 may regard this as the instantaneous plane of the planet's orbit. The disturbing 

 and the disturbed planet will each have its own instantaneous plane. 



Let us now put : 



u, the longitude of a planet counted from a determinate point in the instantaneous 



plane of its orbit. 

 v, its distance from the node of intersection of its own orbit with that of another 



planet. 



y, the mutual inclination of the two orbits. 

 a, sin | y. 



r, the radius vector of the planet, 

 p, its logarithm. 



jU, the attractive force of the sun upon unit of matter at unit distance. 

 a, the mean distance corresponding to the observed mean motion of the planet, 



determined by the condition 



_ (U(l+7) 



n' 



m and n being as usual the mass and mean motion. 



a a , the value of a corrected for the constants introduced by the perturbations, so 

 that, as in the elliptic motion, we have 



p = log +/(/, e , cr), 

 we shall have in the disturbed motion 



p = log a -\-f (?, e, d) -\- periodic terms only. 



a 15 the mean distance of an outer planet, whether it be a disturbing or disturbed 



planet. 



, the logarithm of a. 



a, the ratio of two mean distances, taken less than unity. 

 H, the perturbative function. 



