SECULAR VARIATIONS OF THE ELEMENTS OF 



[HD[III]l±iiJLii£J = 

 &c. 



fo7TinT8ir 5 T»ii 3 .*ir*^i = 

 E7giHIEI][I3[inJ= 



&c. 



=[H3E3[!3[!3 5 



:[ITo][^][IT3^, 



= f«i°1[°Ta|[M |li,»|, 



= riT||l,4| |4,3|[s7?l, 



\. (15) 



= [lT0l|0,4| |4,S||3,2||2,l|, 



^[^[^[^[fT], 



(16) 



4. The quantities (o,i), (0,2), (0,3), &c, (1,0), (1,2), (1,3), &c; |J7T], [5T|], \TJ], Sec, 



pTo] , |i,2| , |i,3| , &c. ; depend on the masses and mean distances of the different 



planets. The analytical expressions of (0,1) and |o,i| are as follows: Mec. Gel. 



[1076], [1082], Bowditch's Translation. 



, . Zm'na?W}? 



(0,1) = 



E3 = 



4(1— a 2 ) 2 ' 

 3m'na \ (1 +a 2 )VV+ h* h -i 1 



(17) 



(18) 



2(1— a 2 ) 2 



In these equations n denotes the mean motion of the disturbed planet ; m! the 



mass of the disturbing planet ; a the ratio of the mean distances of the inner and 



outer planets, &i 0) and b<V depend entirely on a, and are given by equations [989] 



Mec. Gel. If we reduce the coefficients of the different powers of a to decimal 



numbers we shall have the following equations to determine ¥V and 5iV. 



1 ¥V = 1 + 0.25a 2 + 0.015625a 4 + 0.003906249a 6 + 0.001525878a 8 



+ 0.0007476805a 10 + 0.0004205703a 12 + 0.0002596378a 14 



-f 0.0001714015a 16 + 0.0001190288a 18 + 0.00008599836a 20 



-f 0.00006414336a 22 + 0.00004910978a 21 + 0.00003843058a 2 



4- 0.00003063663a 28 -)- 0.00002481567a 30 4" &c. 



(18) 



¥}?=— a + 



(19) 



■ a -f- 0.125a 3 + 0.015625a 6 + 0.004882812a 7 + 0.002136230a 9 

 4- 0.001121521a 11 + 0.0006608960a 13 + 0.0004219114a 15 

 4- 0.0002856691a 17 + 0.0002023490a 19 + 0.0001485425a 21 

 4- 0.0001122509a 23 4- 0.00008688650a 25 + 0.00006862602a 27 

 4- 0.00005514591a 29 4- 0.00004497838a 31 4- &c. 



If we now multiply equation (19) by 1 4 a 2 , and equation (18) by a, and put 



\(l+a 2 )h$+±abW\a=V ), (20) 



We shall have 



Z><°> = — 0.625a 4 + 0.15625a 6 + 0.024414061a 8 + 0.008544920a 10 

 + 0.004005431a 12 + 0.002202987a 14 + 0.0013424452a 16 

 + 0.0008789820a 18 + 0.0006070469a 20 + 0.0004368899a 22 \ . (21) 



+ 0.0003249368a 24 + 0.0002482472a 26 + 0.0001939431a 28 

 + 0.0001 5440856a 30 + 0.00012493996a 32 + &c. 



