THE ORBITS OF THE EIGHT PRINCIPAL PLANETS. 



It will manifestly be unnecessary to compute b { y separately, since it is already 

 included in the value of b m . Therefore taking logarithmic coefficients of equations 

 (19) and (21), we shall have the two following working formulae for the computation 

 of Z><°> and &<*>. 

 Z/ u >= — a ^j 0.625— [9.1938200]/ — [8.3876400],/ — [7.931 7080]r/ 



— [7.6026493]a 8 — [7.34301 19]a 10 — [7.1278964]/ 2 



— [6.9439800]a M — [6.7832222]a 16 — [6.6403720]a u \ . (22) 



— [6.5117989]-/ — [6.3948844J/ 2 — [6.2876743]a 24 



— [6.1886714]x 2 ° — [6.0967014] / 8 — &c. | 



b[V = — a \ 1 — [9.0969100]/ — [8. 1938200}/ — [7.6886700]a 6 



— [7.3296480]a 8 — [7.0498073]/ — [6.8201332]/ 2 



— [6.625221 2]/ 4 — [6.4558633]/° — [6.306 1010]/ 8 



— [6.1718509]/° — [6.050 1898]/ 2 — [5.9389523]a 24 



— [5.8364888]/° — [5.7415133]/ 8 — [5.6530038]/°— &c. 



Then we shall have 



(0 ' 1) = -4(i-/r (24) 



L — i 2(1 — a 2 ) 2 ' { } 



5. To reduce the preceding formulae to numbers we shall assume the following 

 values of the invariable elements of the eight principal planets. 



Invariable Elements of the Eight Principal Planets. 









Mean motions 



Mean distances from 







Masses. 



in a Julian year. 



the sun. 



Mercury .... 



m = 



4865751 



n =5381016".2 



a =0.3870987 



Venus .... 



in' = 



1 + / 

 390000 



n' = 2106641 .438 



a' =0.7233323 



The Earth 



„ 



l+/<" 



»" = 1295977 .440 

 n'" = 689050 .9023 



a" = 1.0000000 

 «'" = 1.5236878 



Mars 



m'" = 



368689 



l + ,r 

 2680637 



Jupiter .... 



m !r = 



1 + """ 

 1047.879 



n ,v = 109256.719 



«" = 5.202798 



Saturn .... 



m r = 



3501.6 



n T = 43996 .127 



a" = 9.538852 



Uranus .... 



m n = 



l+p w 



24905 



n r, = 15424.5094 



a" =19.183581 



Neptune .... 



m r! ' = 



1 + //" 

 18780 



n y "= 7873 .993 



«"'= 30.03386 



