TliE ORB IT OF URANUS. 



C HATTER V. 



COLLECTION AND TRANSFORMATION OP TUE PRECEDING PERTURBATIONS 



OF URANUS. 



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

 tli>turl>in i i,' planets, nor depend on the secular variations of the eccentricity and 

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

 elements. These terms arc as 



(1) In the longitude of Uranus 



11 of Jupiter, -f,31. 21 IT,/ -(-25.657 sin g 



Artioii of Sutiirn, 4-10.9<>9(M +8.545sin<7 



:i of NVptune, 0.42G2I -f O.C97 sin g 



Total, 4-41.7544* +:54.S99nin<7 



J) Iii the value of cos 



-\- 1.397 sin Zg 1.859 cos 9 0.087 cos 2p 



+0. 461 sin 2? 4. 735 cos? O.lfi9cos2<7 



-(-0.04G sin 2y 0.088 COB 3 005 cog 2? 



-{-1.904 sin 2? 6. 682 cos g -- 0.2Glcos2<r 



\ Hi of Jupiter, 10089 



>n of Saturn, 3543 



Action of Neptune, + 138 



Total, 13494 



> units of 7th place of decimals. 



2 sin g -f 1 sin 2g 



losing 

 4- 1 sin </ 



16 sin g -\-l sin 2</ 



492 cos g 33cos2</ 



I.s4rosgr 15cos2<7 



Icoag 1 cos 2g 



G77cosjr 49cos2</ 



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

 ro-nrdinate. If we suppose a change of 6 in the mean motion of a planet, the 

 rorre-ponding change in fy> will be 



dp = 5 . 

 3 n 



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

 eh:mge in fy> will be 18045, and in cos 4-fy>, 18025. Subtracting these from 

 the above perturbations, the secular term in the mean motion will disappear, and 

 we shall have for the constant term of cos 



+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 differences of the values of the secular terms, found by 

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

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

 the other excluded. 



