2fi 



J tlie inclination of the eqnatoi-ial plane to the orliital plane, then 

 we have ; 



z =z — «1 (1 + ^) •^**' (^ + ^') "^^^ J 



? = «J* (1 -h §)^ sm' (r -r V') •^''*'^ ^• 

 For the ealeuiation of the secular portion of the pertnrhative 

 function we thus need the secular portions of 5/', 5/' ,9m- (??-(- tf?) and 

 ,s/,i4 (i^ _j_ ij^,j for different values of />. I get (denoting the secular 

 portion l>v the letter S) : 



* 2 



S%' 



SP = 



Sc* = 



1/3 1 A 



Ssin' (^ +'P) = 2~[s'^ +J^ 'J '^' ^ "^ 



1/3 A 1 



*S è sin' {v -\- \p) = ~ e' { 1 — -^ cos 2 ip + T- é.' cos 2 ip 

 4 V^ 2 /lb 



;S s« siti' {v -\- tp) = e' 



S i' sin- {v -^ \p) = e' 



3 

 S sin' {v 4- If') = - . 



16 



3 

 16 



cos 2 \^ 

 cos 2 \p 



Substituting in the expression for i2 we find : 



-S <i 



C, 



3 



«1' ^2 -^ - ( 1 — :7«i' Q 



1 





16 



-1 sin' J 



1 



1 13 3 3 ^ 

 C, a.'jj' -{-e' \ V — 'tC, a,'p' + 



,73 5 



-i- cos 2 V' I ~ 8 "^ 8" ''' "^ 4 ^^ "''^'' 



+ 



9 45 



64 



105 /I 13 25 , 35 . 



7' + cos2il) —-\ y y' +7^ 7' 



64 '^ ^ ^ I 16 32^ 16^ 32^ 



3 [a' — c') 



16 



— sin'* J. 



Let /, w and ^ l)e the inclination, the longitude of the perihelion 

 and the longitude of the ascending node of the orbital plane of the 

 planet, Jo ^"d *I> the inclination and the longitude of the node of 

 the equatorial plane of the ellipsoid all with reference to a tixed 

 fundamental plane, e.g. the ecliptic of a certain epoch ; then we have : 



sin J cos (tp — to -f^) =: — cos J„ sin i -\- sin J„ cos i cos (^ — *P) 



sin J sin (ip — w -j- ^) = sin (^ — 0) sin J^. 



