THE ORBITS OF THE EIGHT PRINCIPAL PLANETS. 



Ill 



the root g b , wc get, 







&=— 0".661666, 







N, = + 1.232212.V s ' r 



log 



0.0906854, 



N 6 ' =+1.131314JV 5 /7 



" 



0.0535832, 



N'' =+1.108223^/' 



it 



0.0446270, 



Nf =+1.049477.V/ r 



it 



0.0209706, 



NJ = +().9653287iV r i ' r 



t( 



9.9846752, 



iV 5 r/ = — 0.9379252JV/* 



" 



9.9721682?* 



N™=— 9.829270^" 



a 



0.992521 2» 



For the root g e , we get, 



g 6 =— 2".916082, 



^ 



=+ 3.557327.Y /r 



#0 



= + 2.059 163.V/ r 



#• 



=+ l.S45317iV/ r 



W 



= + 1.3 141 87 JV/' 



iV 6 " 



=-f 0.8164588JV a J 



JV 6 " 



=— 20.11 300iV/ r 



iV/' 



'=+ 2.161663iV 6 " 



For the root g-, wc get, 



g 1 —~ 25".934567, 



N 7 =— 0.04207851iV/' 

 iVV =—0.04653315^/' 

 JV/ =— 0.4328950iV/ r 

 2V/" = — 1.468114iV/ r ' 



iV/ =— 2.4907()9iV/ r 

 iV/' =+0. 1093424 JV 7 " 

 jV 7 ™=-f0.qi225277iV/' 



lo£ 



0.5511238, 



0.3136906, 



0.2660709, 



0.1186570, 



9.9119342, 



1.3034768n, 



0.3347880. 



log. 8.6240604?*, 



" 8.6677625??, 



" 9.6363826», 



" 0.1667598m, 



" 0.3963230;?, 



" 9.0387886, 



" 8.0882343. 



5. Having thus determined all the roots of the equation of the eighth degree, 

 together with the ratios of the constant quantities N', iV", N"\ &c, corresponding 

 to each root, the complete integrals of equations (E) will be 



q =N cos (/7'+;3)+^Vi cos (#<+&)+# cos (&*+ &)+&c, 

 </=.¥' cos (^+,,3)+^' cos fat+pj+NJcoa (#<+&)+&*, 

 ? "=W"cos (^4-/3)-j-iV 1 "cos ( <7 1 <+^ 1 )+ iV 2" C0S ( </V+/? 2 )+&c, 

 &c; 



p =N sin (^+/3)+iV 1 sin (at+pJ+Nt si » (&*+&)+&c., 

 p'=iVsin (^+^)+iV;'sin (?/ 1 <-j-/3 1 )4-A T 2 'sin (^+&)+&c, 

 y^iV'sin (^+,.3)+iV;*sin ( ? 7l <+ l ^ 1 )+iV 2 "sin (0+&)-\-&c, 

 &c. 



The analysis of § 16 will conduct us to the following equations for the deter- 

 mination of the arbitrary constants corresponding to each root. 



