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, g u //,, &c., N", N{, JV„", &c., (3, /?„ (3 2 , &c., cor- 

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

 — the numbers in brackets being logarithms, 



23° 27' 31",0— h [ 8 - 7069903 3 [8.4509983] [8.67151 l(i] 

 l+g l+9i l+g* 



[9.1494 996] [6.9157253] [7.5163249] 



+ i+g, " i+g> + i+g, 



[8.6222025] 



4509982] 



■ (555) 



50".235724-0".05933222cot7 t =7 + { [8-7069903] [8.45099 



[8.6715146] [9.1494996] [6.9157253] 



~l+92 l+9s + i+g* 



[7.5163249] _ [8.6222025] K tanA 



+ 



. (556) 



If we divide equation (556) by I tan 7/, and add the quotient to equation (555), 



we shall get 



50".23572+0".05933222cot7t , OQ0 ~, ,„ ft . . . , ,-,- 



L U23° 2 1 31.0=7t4-cot h. (o5 



Ztan7t ' ] v ' 



Whence we get 



50".23572-{-0".05933222 cot h 55g . 



l+(7i— 84451".())tan7i ' C ° 



The direct determination of h and I 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 h. If we sub- 

 stitute 7t=23° 17' 16" in equation (558), we shall get 7=50".4382997; and if we 

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



7t=23° 27' 31".0— 10' 14".4265=23° 17' 16".5735. 



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



?=50".4382387. 



Having found h and ?, 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, g u g„, &c. to 7, we shall get/, /„ /,, &c, as follows, 



/=45".312168, A=l =50".438239, 



/,=43 .846111, / 5 =49 .776573, 



/ 2 =33 .044849, / 6 =47 .522157, 



/ 8 =32 .029325, / 7 =24 .503672. 



