THE ORBIT OF URANUS. 95 



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

 found by Ohifsen, but I cannot reproduce this 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 Lo Vcrrier are {Aauides <le VObscrva- 

 toire Ivijjcrud de Paris, tome ii, p. 101), 



// // ff If 



Secular change = — 47.59 — n.52j' — 28.90j/ — 0.83j'"' 

 Mot. at equinox = + 5.89 -f 0.62r + IMi' + 0.73r"'. 



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



POOTOITO' 4OI78I7' '""'^ 2;680;337' respectively. The influence of admissible 

 changes in the masses of the other plants is insensible. 



From the researclics of Lo Verrier on the motions of the four inner planets I 

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

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

 Hansen, namely, 



These values give for the motion at tlie 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, 



5,9 = (5".437'+ 0".19 T") cos v + (4G".78r— 0".0G 7"^ sin v. 



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

 we find 



hii = J(10".12 + 1".14./) T-{- 0".19 T-\ cosv 

 + |(41".54 — 0".52«) 7'— 0".0G r\ 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 



2) ^ sin (|) sin 0, 

 q = sin ^ cos d ; 



■we have 



sin ^ = — p cos V -\- q sin v 

 cos I38i3 = — ^;p cos V -|- bq sin v. 



