THE ORBITS OP THE EIGHT PRINCIPAL PLANETS. 167 



3. If we neglect the squares of e, e', e", &c., d>, d>', d>", &c., y, we may put 

 cos y=l; sin y=tan y=y; sin d>=tan d>; sin $'=tan d>', &c.; 



then we shall have 



psin $ sin 6, /=sin d>' sin 6', p"=sin d>" sin 6" &c., ) - ,. 

 5=sin d> cos 0, </=sin d>' cos 6', </'=sin d>" cos 0" &c. j 



Substituting these values in equations (408) and (409), also putting ;=0, and 

 remembering that for the root # 4 =0, we have N t =N^=N t v , &c., we shall have 



m 

 na 



m 



sn <> sn 



na na 



5. Jtf 4 sin/3 4 ; 



^sin d>' sin 6'+ &c.= I 

 na { 



sin d> cos 0+-' sin d>' cos 6'+ &c.= I ^-. 4-&c. 1 N t cos #,. 

 Jza' I na ' 'a ' j 



na na I na ' n'a' 



But if we neglect m 2 , m' 2 , &c., we shall have 



(535) 



in 



i i m m 



mna~= , m''a-=- , mnV= --- , &c., 



"" 



a 



and equations (527) will give, by substituting the values of c,c',c", 



A i wi' . . . , , o \ m in' , n } 



sin 04 sin d> sin 4-&c.== < _L_&C. > y sin II: 



' na [ na ' nrf ' J 



i 04- m t , sin d>' cos 0'4- &c. \ U-'~, 4-&c. i y cos II. 

 ' na ( na ' na } 



m 

 sin d> sm 

 na 



-sm d> cos! 



(536) 



Comparing equations (535) and (536) we find n=/3 4 , and y=N. Now substi- 

 tuting n=/3 4 , y=JV 4 , in equations (532), they will give 



sin d> sin =sin <(> sin (0 /3 4 )=sin d> sin cos j3 4 sin <> cos sin /3 4 



=p cos /3 4 ? sin /2 4 => ; 

 sin ^ cos =sin(|) cos (0 /3 4 ) y=: 

 sin d> cos cos/? 4 sind) sin0sin/3 4 y =qcos(3 t ^)sin/3 4 N 4 =q . 



(537) 



And since the relative values of N, N', N", &c., N^ N lt N^", &c., are known we 

 may determine their actual values corresponding to the invariable plane, by the 

 analysis of Chapter II, 5. We shall therefore suppose 



?0 =a N cos (grt+^O+a! ^ cos ( g r 1 <+/3 1 (0) )+a 2 ^ 2 cos 



9o '=a ^' cos ( fl r+ i 8TO)+a 1 ^' cos (9' 1 <+/? 1 (0) J+a2 ^' cos 



&c.; 



Po =a N sin (^-f/^^+a! ^ sin (gj+pW+a, N, sin (^<4-/8 s (0 >)+&c., 



p '=a N 1 sin (^+/30 > ))-[-a 1 JV,' sin 07 1 <+/3 1 (0) )+a 2 JV 2 ' sin 



&c., 



(538) 



a, HI, a 2 , &c., being the constant factors which are necessary in order to reduce 

 the numbers already calculated to the corresponding ones for the invariable plane ; 

 and /3 (0) , (3^\ 8 2 (0 \ &c., being the constants necessary to satisfy the equations for 

 the given epoch. 



