THE OK 13 IT OF URANUS. 95 



Ilansen mentions 5".39 as the corresponding motion at the equinox of 1850, 

 found by Olutsen, but I cannot reproduce tliis result from the secular diminution 

 with any masses of Mercury, Venus, and Mars, which seem to me probable. The 

 expressions in terms of the masses given by Le Verrier are (Annules de VObaerva- 

 tuire Iiiijicritil </e J'aria, tome ii, p. 101), 



// a n 



Secular change = 47.59 0.52p 28.90j/ 0.83i>" 

 Mot. at equinox = -f 5.89 -f 0.62* -4- 7.57i/ -f 0.13/*. 



In this expression the masses of Mercury, Venus, and Mars are represented by 



\-\-v 1 + v' 1-f-v" 



a,0<)0,0i).r 401,847' a " l 2,680,337' "V^^- e influence of admissible 

 changes in the nuisses of the other plants is insensible. 



From the researches of Le Verrier on the motions of the four inner planets I 

 conclude that the following arc about the most probable distribution of the correc- 

 tions of the masses necessary to produce the motion of the obliquity given by 

 llansen, namely, 



v = | 

 v' = .018 



These values give for the motion at the equinox of 1850 



+ 5".43 



Introducing the secular variation of these motions we have, for the change in 

 the latitude of any celestial body near the ecliptic, arising from motion of the 

 ecliptic, 



&(3 = (5'A3T+ O'.ig? 77 ) cos v + (46".78T 0".06 7") sin v. 



Combining this with the change arising from the motion of the orbit of Uranus, 

 we find 



W = f (10M2 + lM4ji) T+ OM9 T*\ cosv 

 + |(41".54 0".52//) T 0-.06 T j sin v. 



We may represent these expressions in the usual way by secular variations of 

 the inclination and node of Uranus. But, owing to the small inclination, and 

 consequent rapid motion of the node, it will be necessary to include the coefficients 

 of the second power of the time. On the other hand, no distinction between r 

 and is necessary. Putting $ for the inclination of the orbit, for the longitude 

 of the node referred to the equinox of 1850, and 



p = sin <f> sin 0, 

 q = sin <p cos0; 



we have 



sin = p cos v + q sin t> 

 cos {3&{3 = ftp cos v -j- 5 sin v. 



