THE ORBIT OF URANUS. IQ 



CHAPTER V. 



COLLECTION AND TRANSFORMATION OF THE PRECEDING PERTURBATIONS 



OF URANUS. 



The terms of the perturbations which neither contain the elements of the 

 disturbing planets, nor depend on the secular variations of the eccentricity and 

 perihelion, admit of being greatly simplilied by a slight change in the arbitrary 

 elements. These terms are as follows: 



(1) In the longitude of Uranus 



Let tis first consider the first or constant term in the perturbation of each 

 co-ordinate. If we suppose a change of hn in the mean motion of a planet, the 

 corresponding change in ho will be 



' 3 u ' 



If, then, we increase the mean motion of Uranus by 41".754, the corresponding 

 change in hp will be — 18045, and in cos -J/^p, — 18025. Subtracting these from 

 the above pcrturl)ations, the secular term in the mean motion will disappear, and 

 we shall have for tlie constant term of cos ^l^Sp 



+4531 



This same change in the mean motion will produce a secular term in the equa- 

 tion of the centre of the same nature with that produced by the secular variation 

 of the perihelion. The diff"erences of the values of the secular terms, found by 

 the two methods employed in Chapters II. and HI., proceeds from the fact that in 

 the one case the effect of the above term in the mean motion is included, and in 

 the other excluded. 



