Rg (<?)-(/)] THE SATURNIAN SYSTEM-. ( J 



N = V Tr sm <p -4- u — A) I = 2 /• cos j 9 - „ 



L fix 'J J 0(0) 



= 27" C»* + 2 ( g - *fl F + ( s ~ *) 3 - <? r cos »] 

 (^ = ^3 (2/20 — »*) (/» — 9 r cos cp). 



Lastly, to find P, we may neglect the terms which do not affect its value, which 

 gives the reduced values 



/ 2 (r _ () cosc f ) = 2r^cos 2 T — [3;^ + V 3 + (»(,= — i) 2 ] cos < r + ^ + ro 2 + r (. ? — ^) 2 , 

 — (a — r cos 9) r sin 2 9 = — r 2 cos 3 9 -(- r (> 2 cos 2 9 -j- r 2 9 cos 9 — <r r, 

 j* ^. — ^ cos 9) — (0 — r cos 9) r 9 sin 2 9 = — r 2 < s > cos 3 9 -{- 3 r 2 cos 2 9 



- [2 ^4- (,3+0(3 — J) 2 ] cos 9, 



J r cos 29 [/l(r — cos 9) — (0 — r cos 9) r^ sin 2 9] = — r 3 (,> cos 5 9 + r 2 ^) 2 cos 4 9 



— [I r\> + r v 3 + r (* — £) 2 ] cos 3 9, 

 [r 2 sin 3 9 + (0 — 5) 2 ]y1 (0) = 2 r- 9 cos 3 9 - r 2 (r 2 + </) cos 2 9 — 2 r [r + (s - bf] cos 9. 



The quotient of the second members of the two last equations gives 



(„) . P = - I cos- 9 + (^ - r J C0S9 + — 2 -f -^ j5-. 



The combination of the equations (a), (b), (c), (d), and (e) gives 



or = r m = - 9 2 X io s (/ 2 + ^ - f ) 







_ f (g-6)(g' + >- i 0(e a -rT , ( -— 6)( e « + p«)(e*-* 3 ) ] 'f 1 



I lG t ) 2 "T" 16 t , 2 J ./^/ 2 /o 2 (0) 



+ X *"X ^ ** cos2 (p + r« (,/ + r2) cos y + ^? ~ 9 r cos f/ ' + '" + ( " ~ bf 



_i^cos 2 9+(^ ? — ^cos^-K-l^-H*- *) 2 ]> 



■fj^^-rtC— flfjsJc-K— »)JT[A- !=± 5r s i 



35 ( 27 '> 



