2(5 THE ORBIT OF URANUS. 



These rotations will be around the same principal axis with the rotations dyj 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 ^57, blc, h^^ 

 and hlc will produce the following changes in v, v', and y. 



^v = cot yhk — coscc yhk' 



bV = cosec yhk — cot yhk ' (29) 



hy = hrf — hvi 

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

 order added to hR will be 



,„ (dR , , dR \„ 



^-^ "" \ «? V "^"^ ^ + cJ V'" ''''^^'' ^ n 



(30) 

 /dR , dR . \,,. 



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



{ ^ (^v^ + "^v^) ^'"'^'''' r + cot y) - 3 (^^ - ^^ ) (cosec y - cot y) J hk 



~ { '' (4v" + ~w) ^''"''''' y + c«t 7) + ^ {~sy ev) ^'""'^'^ ^ ~ """^ ^^ } ^^' 



But, 



cosec y -[- cot y = cot ^y = ^-^• 



coscc y — coty = tan ky = , — . 



^ ^ ^' cos 1 y 



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



dR m'h ,. . ... „ 



^ = ---(»+^)smi\r 



dR m'h ,., . .,. . „ 



_, = ---(. +.;)-niV^. 



Making these substitutions, and putting, as before, 



the above value of hR reduces to 



5E = "- I < cot 1 y {hk - hk') + (I +y -i' -f ) tan i y (5^ + ^^'O \ sin i\^ 



, m' cosh y dh ,^ , ^ ^ at ("^l) 



' 2ai o'er 



