OF URANUS AND NEPTUNE. Vll 



m 



+ 

 4 



+ 

 4 



m 

 4 



m 



+ 

 4 



- tut f (2ft - 1) B k + a ^4 e sin"£ 7 ' cos {ft(£ - f) - 0- £ + £- 2n') J 

 li (_ dd J 



^aa'|(2& + 3) J B ft + a^Je S in 2 l 7 'cos{A(e-f)-/3 + (e+r-2n')} 

 _^.'a«'|(8Jfc-l)B fc + a^Jeain»i 7 'co8K*-l)(?-r)-j8} 



- *' aa' {(8ft + 3) 5, + «^H « ■&*$?' cos H* +»)(£■- D " #} 



- ™' oo' {(2ft + 1) j? s - a' ^} e' sin'| 7 ' cos {ft (£ - f) - # - £ + f - flT) } 



^aa'{(2ft-3) J B,-a'^} C 'sin^ 7 'cos^(?-n-/3' + (e+r-2n)f 

 l a'|(2ft-3)^-a'^J e 'sin»l 7 'co S {(ft-l)(^-f)-/3'} 



+ i aa'{(2ft+ l)i? A -a'^Je' sin 2 l 7 'cos{(ft + 1) £ - §') - /3'}, 

 in which positive and negative values of k are to be taken, including zero. 



8. We shall find in the developement of R the following terms of the first order in 

 eccentricities which involve (n — 2n) t, putting \ = nt + e — 2 (n't + e'), 



i?! = m'Mye cos (\ + "&) 

 + m'M 2 e cos (X + ■&'), 



where aM x = - (4& 2 (£) + a — - — ) , 

 2 V da J 



*, a f i m d6 ' Q> \ 



aJfj = 3&,W - 4a + a — ; — , 



2 \ da J 



a being = -j , and 6,(1) the coefficient of cos ft (0 - 0') in the expansion of 

 a 



{l -2acos(0-0') + a 2 }-*. 

 Terms of the third order involving X, are 



R = m (M 3 ^ + M^e' 2 + M b e sin 2 l 7 ') cos (X + sr) 

 + m (M 6 e' 3 + Af 7 eW + M 8 e'sin^y) cos (X + sr') 

 + m'M s eV cos (X + 9. w - •zzr') 

 + tri M io ee' 2 cos (X + 23r' - nr) 

 + m'M n e sin 2 l 7 ' cos (X + 2 II' - ■&■) 

 + m'M n e sin s l 7 ' cos (X + 2 II' - "&'), 



