THE ORBITS OF THE EIGHT PRINCIPAL PLANETS. 167 



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

 cosy=l; siny=tany=y; sin <|>=tan <p; sin <£>'=tan <p', &c; 

 then we shall have 



p=sin $ sin 0, jt/=sin <|>' sin 0', p"=sin <|>" sin 6" &c., 



:£;}(584) 



§=sin ^ cos 6, j'=sin <?>' cos 0', (/'—sin <|>" cos 0' 



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

 remembering that for the root #.,=0, we have ^=27^=^ t &c, we shall have 



m 



sm 



na 



in $ sin 0_j_ ™ sin d/ sin 0'+ &c.= { — - +-"-+&c. i iV 4 sin /3 4 ; 

 na { na ' na ' j 



in <£ cos 0+-^_sin <2>' cos 0'+ &c.= j —+— +&c. 1 N t cos ft. 

 «a' ( na ' w'a ' j 



(535) 



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



ml 



mna'— , mna~z 



na n'a' 



ft ft If) *ll> n 



\ n a ■=.—, -„, &c, 



n a 



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



sm d> sin 0J — — sm d> sm 4- &c.= <^ U&c. ysinll; 



«a ' raa ( na n'a ' j 



?n 



smd>cos0J sin*cos04-&c.= ^ U- -U&c. V y cos n. 



na na (. rea ' wa ' j 



(536) 



Comparing equations (535) and (536) we find II=/? 4 , and y=iV 4 . Now substi- 

 tuting II=/? 4 , y=iV 4 , in equations (532), they will give 



sin <2> sin o =sin cp sin (0 — ft)=sin <p sin cos /? 4 — sin 4) cos sin /? 4 



=p cos (3 1 — q sin (3 i =p ; 

 sin<|> o cos0 o =sintf)cos (0 — /3 4 ) — y= 

 sin <|> cos cos /3 4 — sin <p sin sin /? 4 — y =<? cos /3 4 — p sin /3 4 — iV 4 =5' . 



(537) 



And since the relative values of N, N', N", &c, iV,, N lt iV/', &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 



q =a N cos (gt+pmy+o^ N, cos ^+(3™)+** ^2 cos Gfc<+&<°>)+&c., 1 



?0 '=a N' cos (gt+ t 3 (0) )+ai W cos (g^ft^-fa, JV 2 ' cos (^+& (0 >)+&a, 



&C; 



#, =a N sin (^+/3«»)-|- ai ^ sin (gj+fcvn)^ N, sin (flr 2 *+&< >)+&c, 



i> '=a 1ST sin (< 7 <+ / 3«")+a 1 i^' sin (i/ l ^+/? 1 (0) )+cc 2 A? sin (<fc*+& (0) )+&c., 



&c, 



a, a l5 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) , /V', {3 2 m , &c, being the constants necessary to satisfy the equations for 

 the given epoch. 



(538) 



