SECULAR VARIATIONS OF THE ELEMENTS OF 



(15) 





H] [Hil [HI] HH] [HiD H3 [HE GUI] 



ZEl Hill EH] E3 ~ E3 E3 EH] UH] > 





0^1|l.8||.8||g.4l[4.o| = |l.Q||o.4||4.S||3.a||a.l|, 



0,1 1,2 a,* h,3 3,0 = 1,0 0,3 3,4 4,2 2,1 , 



1,2 2,4 4,3 3,5 3,1 = 2,1 1,6 5,3 3,4 4,2 I 



4. The quantities (0,1), (0,2), (0,3), &c., (1,0), (1,2), (1,3), &c.; [uj], FT^l. 1^1, &c., 

 [TTo], [T7T|, QTT], &c. ; depend on the masses and mean distances of the different 

 planets. The analytical expressions of (<M) and [ , ' | are as follows: Mec. Gel. 

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



(18) 



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, b ( $ and b ( ^ depend entirely on a, and are given by equations [989] 



Mec. C61. If we reduce the coefficients of the different powers of a to decimal 



numbers we shall have the following equations to determine b ( $ and b ( } k \ 



1 6i) = 1 + 0.25a 2 + 0.015625a 4 -f 0.003906249a 6 + 0.001525878a 8 



-f- 0.0007476805a 10 -f 0.0004205703a u + 0.0002596378a w 



+ 0.0001714015a 16 + 0.0001 190288a 18 4- O.OOOOSSggSSGa 20 



4- 0.00006414336a 22 + 0.00004910978a M -f- 0.00003843058a 26 



-j- O.OOOOSOGSGGSa 28 -j- 0.00002481567a 80 4- &c. 



ilV= a + 0.125a 3 + 0.015625a 8 + 0. 0048828 12a 7 + 0.002136230a 9 

 4- 0.001 121521a u + 0.0006608960a 13 + 0.0004219114a 15 

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

 -j- 0.0001 122509a 23 4- O.OOOOSGSSGSOa 28 + 0.00006862602a 27 

 + 0.00005514591 a 29 + 0.00004497838a 81 + &c. 

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



. | (l+a 2 )6[V+|a#. ( J ) | =& (0) , (20) 



we shall have 



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



+ 0.004005431a 12 + 0.002202987a 14 + 0.001 3424452a M 

 \ + 0.0008789820^ + 0.00()6070469a 20 + 0.0004368899a 23 

 4- 0.0003249368a 24 + 0.00024824720, 26 + 0.0001 939431a 28 

 4- 0.0001 544085601" -j- 0.0001 2493996a s2 + &c. 



(19) 



(21) 



