166 



SECULAR VARIATIONS OF THE ELEMENTS OF 



Substituting in these equations the values of m, n, a, e, q> and d, given in §§ 5 

 and 17 of Chapter I, and § 6 of Chapter II, we shall get 



c=+0.0035274157, c'=— 0.00002735230, c"=+0.00009393304. (530) 



In finding these quantities n" has been supposed to equal unity, and the values 



of n, n', n'", &c. have been multiplied by — - in order to preserve the same ratio. 



n 



Substituting the values of c, c', and c", in equations (527), we shall obtain 

 n=106° 14' 6".00, and y=\° 35' 19".376. (531) 



If we now denote by <|> , <p ', ^> ", &c., 6 , $ ', 6 ", &c, the respective inclinations 

 and longitudes of ascending nodes of the different planets, on the invariable plane ; 

 the values of 6 , O ', 6 ", &c. being reckoned from the descending node of the fixed 

 ecliptic of 1850, on the invariable plane; we shall have the following equations to 

 determine 6 , 6 ', &c, ^> , <£>„', &c. 



sin <|> sin o =sin <p sin (6 — II), 1 



sin <p cos o =cos y sin fy cos (6 — II) — sin y cos q>. J 



These equations will give the following elements: — 



(532) 



Mercury, 



<Pa =6° 



20' 



58".08, 6 =287° 



54' 



5". 12, 



Venus, 



Qo =2 



11 



13.57, O ' =307 



14 



8.10, 



The Earth 



, <?>o" =1 



35 



19.376, O " =180 







0.00, 



Mars, 



4>o" =1 



40 



43.70, O '" =248 



56 



21.45, 



Jupiter, 



<bT=0 



19 



59.674, O ^ =210 



7 



35.44, 



Saturn, 



?o r =0 



55 



30.924, 6 r = 16 



34 



26.66, 



Uranus, 



fc"=l 



1 



45.27, O F7 =2O4 



12 



33.78, 



Neptune, 



4>o™=0 



43 



24.845, o y/J =286 



39 



55.10. 



2. Now putting 













tan q> Q sin 6 - 



=Po, 



tan <2>' sin O '=^ O ' &c, ) 

 tan <p' cos 8 '=q ' &c. j 



(53 



tan (£> cos 6 Q - 



=qo, 





we shall get the folio 1 



tving values 









Po = 



=— 0.1058879, 



q =+0.0342038, 





Po = 



=—0.0304057, 



q ' =+0.0231090, 





p " = 



-- 0. 





q " =—0.027' 



7354, 





K = 



=—0.0273512, 



q Q m =—0.0105324, 





Pa = 



:— 0.0027 



3067, 



q IV =—0.00503058, 



Po V = 



--+0.00460691.. 



q r =+0.0154792, 





Y> VI — 



Pa — 



=—0.00736719, 



q rT =— 0.0163856, 





Pa — 



-.+0.0126079, 



£ ™=+0.000734628. 



If we substitute these values in equations (408) and (409), we shall obtain the 

 values of @, /3 15 /3 2 , &c., N, N lf N 2 , &c, corresponding to the invariable plane. But 

 instead of performing this operation separately for each root, we shall proceed in 

 the following manner. 



