OF URANUS AND NEPTUNE. IX 



The terms of the third order involving 3\ are 

 R 3 = m'K^e 3 cos (3\ + 3sr) 

 + m'K^e' 3 cos (3\ + Sur') 

 + m'K 3 e-e' cos (3\ + 2-sr + ■&■') 

 + tn'Kiee" 2 cos (3\ + sr + 2sr') 

 + m'K b e sin 2 !^' cos (3\ + 211' + w) 

 + m'K 6 e sin*!'/ cos (3\ + 211' + ■&'), 

 where 



a^-, = — J 804& 6 (}) + 279a — f— + 30a 2 — ^— + a 3 — V > , 

 48 [ da da~ da 3 J 



aK 2 = |8166ji<1) + 312a -4— + 33a 2 — V + « 3 — ^-> » 



48 I da da 2 da 3 ] 



ojr » = - 77; |8256 5 (i) + 291a — f- + 31a* -^- + a 3 ~^~) , 

 16 [ da da 2 da 3 J 



aJT 4 = — ^8826/1) + 302a — i— + 32a 2 — — - + a 3 — -±— V , 

 16 I da da 2 da 3 ) 



9. The relation existing between the above coefficients and the corresponding coefficients 

 for Neptune is, for those terms which arise from the symmetrical parts of R and R', very 

 simple. If m'M be the coefficient of any term in the symmetrical part of R, and rnM' the 

 corresponding coefficient in R\ then M = M'. And there also exists a very simple relation 

 between the corresponding coefficients in the unsymmetrical parts of R and R', which may be 

 proved in the following manner. Considering only the unsymmetrical part, 



_ . xx + ««' + **' ra ( dV dry dV\ 



R = m' ?L = -- ; *— +y-J- + « — ), 



r 3 l +m \ dt- dt air I 



x l d"x' d~x' 



since — = — — . &c, the differential coefficients — — , &c. being taken as if the ele- 



r 3 l + m dt' dt~ 



ments were constant. 

 Similarly, 



, drx , dry , drx\ 

 *)- 

 Now any term in this part of R involving (kn ± k'ri) t in its argument can arise only from 



i • i • , . , . , • <&">' d°y , dV . , 



terms in x, y, and % involving lent combined with terms in — — ■ , —— , and — — respectively, 



CLZ GLv (IZ 



involving k'n't in their arguments ; and similarly, with respect to the terms of that form 

 in R". 



Let, then, p cos (knt + q) be a term in x having in its coefficient some particular power of 



[B] 



_, ml, drx , dry , drx\ 



R =-V^n{°°dI- +y ~df + * d?) 



