166 



SECULAR VARIATIONS OF THE ELEMENTS OF 



Substituting in these equations the values of m, n, a, e, ty and 0, given in 5 

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



c=+0.0035274157, c'= 0.00002735230, e"=+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. 



?i 



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

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



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

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

 the values of , ', ", & 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 , ', &c., <><, <fr>'> &c. 



sin <2> sin =sin <> sin (0 II), ) 



sin $ cos =cos y sin <> cos (0 n) sin y cos $. j 



These equations will give the following elements: 



(532) 



2. Now putting 



tan $ sin 6 =p , 

 tan <> cos =<7 , 



we shall get the following values 



p a =0.1058879, 

 Pa =0.0304057, 

 Po' = 0. 

 pf =0.0273512, 

 p ' v = -0.00273067, 

 Po r =+0.00460691, 

 p"= 0.00736719, 

 Po r "= +0.01 26079, 



tan 

 tan 



, sm = 

 , cos O a =q 



;=?}&c.'} 



(533) 



q =+0.0342038, 

 qj =+0.0231090, 

 q " =-0.0277354, 

 q " =-0.0105324, 

 q ' T =0.00503058, 

 q r =+0.0154792, 

 9o r/ = 0.0163856, 

 ?0 m =+0.000734628. 



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

 values of /3, /?j, /2 2 , &c., N, N^ 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. 



