178 



SECULAR VARIATIONS OP THE ELEMENTS OF 



3. The inclinations of the orbits of the planets and the longitudes of their 

 ascending nodes on the ecliptic, or variable orbit of the earth, are denoted by 

 <pi, <pi\ $i"'s & c -s 0i > 0i'> 0i'"> & c -> anc * nave k een computed by the equations 



tan0 1 =(p— p")-r-(q-— q"); tan6 1 '=(y— p")— (?'— §"); &c; 



tan ^j =(p — p")-=-sin X =(g — 2")-^cos Q x ; 1 

 tan q>{=(p'— p")4-sin $/=((/— </')-=-cos 0/, &c. J 



The longitudes of the perihelia and nodes are in every case counted from the mean 

 equinox of 1850.0. 



4. The precession of the equinoxes on the fixed ecliptic of 1850, and the incli- 

 nation of the equator to the same plane, are denoted by ^ and e x ; and they are 

 determined by equations (550) and (551). Reducing them to numbers, and trans- 

 forming so as to dispense with the arbitrary constant quantities, they will become 



^=^+[4.0317043] sin \ft cos |/*+[3.6828039] sin $/* cos \fj 

 —[3.5715890] sin 2 J/* +[3.8013126] sin 2 |/^ 



+ [3.5080433] sin \U cos \f.t 

 +[3.7464212] sin 2 i/ 2 * 

 —[4.0645663] sin \f 3 t cos. \f 3 t— [3.8713230] sin \f\t cos \fa 

 +[4.6281028] sin 2 |/ 3 * ' —[4.4070533] sin 2 |j# 



+[3.0831687] sin \f h t cos \fj 

 —[2.6561496] sin 2 §/ 5 * 

 —[3.0600136] sin '\fjt cos |/ fl f— [3.5903084] sin \f.t cos \fit 

 —[3.0760078] sin 2 §/ 6 * —[3.7238538] sin 2 !'/;* 



(565) 



£l =23 c 



27' 31". 00+[3. 1962358] s 



n \ft cos 



hft" 



+ [3.6563511] si 



*%ft 





—[3.4230485] s 



n±/ x i!cos 



\ft 



+[3.3045398] s 



&h& 





— [3.3390526] s 



m \f4 cos 



zfst 



+[3.1006747] s 



m%f 2 t 





—[4.2160834] si 



n | f 3 t cos 



\M 



— [3.6525469] s 



*h/J 





+[4.0408748] s 



n J/^cos 



\fi 



—[3.5051445] s 



tf\M 





+[2.2889032] s 



ax \f b t cos 



\te 



+[2.7159223] s 



rf\U 





+ [2.7049217] s 



tn ±f J, cos 



2jst 



—[2.6889275] s 



rfkfj 





+[3.2799396] si 



n ^f.t cos 



2 fit 



—[3.1463942] si 



*%fJ 





(566) 



