26 



THE ORBIT OF URANUS. 



These rotations will be around the same principal axis with the rotations dv\ and 

 dk, but around a secondary axis in the plane of the disturbing orbit, and therefore 

 making an angle y with the secondary axis of the disturbed orbit. A geometrical 

 construction will now show quite simply that the infinitesimal rotations &?, &7c, &/, 

 and (W will produce the following changes in v, V, and y. 



&v = cot y&k cosec y&k 1 



v' = cosec y%k cot y&k' (29) 



If we substitute these values in the general formulae (11) the terms of the second 

 order added to &R will be 



. dR 



(30) 



== 



. dR 

 - cosec / H- - cot 



The first two terms of this expression may be put into the form 

 dR. i SR dR 



co - 



dR 



. 



(cosec / + co 



But, 



QR 



cosecy + cot y = cot | y = 



, 



(cosec 



. 

 - cot y) 



N ) s , 



~ cot y} \ 



cosec y cot y = tan i y = _ 



cos Jy 



and in the general term of R, by (7) 



5^2 m'% , . . ... ,, 



-r = ----- (t + ? ) sm Jv 

 <9v a, v - 



Making these substitutions, and putting, as before, 



the above value of &R reduces to 



SR = ~ \ i cot | y (Sfc MO -f (i + y i' / ) tan 



wj'cosly 8h 



+ j#) | sin 



(31) 



