THE ORBITS OF THE EIGHT PRINCIPAL PLANETS. 173 



The first members of equations (553) and (554) are therefore known; and if we 

 substitute in them the values of g, </ lt r/ 2 , &c., N", Nf, Nf, &c., (3, /? 0,, &c., cor- 

 responding to the assumed masses, together with /,//,, &c., they will become 

 the numbers in brackets being logarithms, 



23 21' 31" h C 8 - 7069903 ] [8.4509982] [8.6715146] 



1+<J l+ffi 1+92 



[9.1494996] [6.9157253] [7.5163249] 



~f+ff7~ 



[8.6222025] 



(555) 



f PR TnfiQQO^I l~8 d^nQQROl 1 



50".23572+0".05933222 coa=Z+ j I J -f L ^ DU 



[8.6715146] [9.1494996] [6.9157253] 



[7.5163249] [8.6222025] \ 



~ 3 



(556) 



If we divide equation (556) by Z tan A, and add the quotient to equation (555), 

 we shall get 



50".23572+0".05933222cotA 3 l-.0=ft+cot h. (557) 



Ztan/t 

 Whence we get 



50".23572+0".05933222 cot Ti 





l-J-(7i_ 84451".0)tan 



The direct determination of h and Z from equations (555) and (556) is trouble- 

 some, and it is better to solve them by approximation. A few trials will show 

 that 23 17' 16"=83836" is a near approximation to the value of li. If we sub- 

 stitute 7i=23 17' 16" in equation (558), we shall get Z=50".4382997; and if we 

 substitute this value of Z in equation (555) we shall find 



7i=23 27' 31".0 10' 14".4265=23 17' 16".5735. 



Now, substituting this value of h in equation (558), we shall get 



Z=50".4382387. 



Having found h and Z, we must substitute them in equations (550-553), and we 

 shall obtain the expressions for the numerical values of the precession and obliquity 

 during all past and future ages. 



Adding^, g lt g z , &c. to Z, we shall get/,/,,/ 2 , &c., as follows, 



/=45".312168, f t =l =50".438239, 



/=43. 846111, / 6 =49 .776573, 



/ 2 =33 .044849, /e=47 .522157, 



/ 3 =32 .029325, / 7 =24 .503672. 



